a r X i v :a s t r o -p h /0303593v 1 26 M a r 2003Mon.Not.R.Astron.Soc.000,1(2001)Printed 2February 2008(MN L A T E X style ?le v2.2)
Numerical Modeling of Gamma Radiation from Galaxy Clusters
Francesco Miniati ?
Max-Planck-Institut f¨u r Astrophysik,Karl-Schwarzschild-Str.1,85740,Garching,Germany 2February 2008ABSTRACT We investigate the spatial and spectral properties of non-thermal emission from clus-ters of galaxies at γ-ray energies.We estimate the radiation ?ux between 10keV and 10TeV due to inverse-Compton (IC)emission,π0-decay and non-thermal bremsstrahlung (NTB)from cosmic-ray (CR)ions and electrons accelerated at cosmic shocks as well as secondary e ±generated in inelastic p-p collisions.We identify two main region of pro-duction of non-thermal radiation,namely the core (also bright in thermal X-ray)and the outskirts region where accretion shocks occur.We ?nd that IC emission from shock accelerated CR electrons dominate the emission at the outer regions of galaxy clusters,provided that at least a fraction of a percent of the shock ram pressure is converted into CR electrons.A clear detection of this component and of its spatial distribution will allow us direct probing of cosmic accretion shocks.In the cluster core,γ-ray emission above 100MeV is dominated by π0-decay mechanism.At lower energies,IC emission from secondary e ±takes over.However,IC emission from shock accelerated electrons projected onto the cluster core will not be negligible.We emphasize the importance of separating the aforementioned emission components for a correct interpretation of the experimental data and outline a strategy for that purpose.Failure in addressing this issue will produce unsound estimates of the intra-cluster magnetic ?eld strength and CR ion content.According to our estimate future space borne and ground based γ-ray facilities should be able to measure the whole non-thermal spectrum both in the cluster core and at its outskirts.The importance of such measurements in advancing our understanding of non-thermal processes in the intra-cluster medium is discussed.
Key words:acceleration of particles —galaxies:clusters:general —gamma rays:
theory —methods:numerical —radiation mechanism:non-thermal —shock waves
1INTRODUCTION
The structure and evolution of the large scale universe is a fundamental tool of cosmological investigation.Clusters of galaxies in particular,being the largest bound objects in the universe,have been studied extensively in order to “weight”the cosmos and to probe its structure.In the currently favored cold dark matter (CDM)hierarchical scenarios structure formation originates from the collapse of primordial density perturbations seeded at an in?ationary epoch (Peebles 1993).In this depiction,rich clusters of galaxies correspond to linear perturbations on scales of order ～10Mpc comoving and,therefore,their potential wells are expected to accommodate a representative fraction of the actual mass content of the universe (White et al.1993).By the same token,the environment there should be the end result of physical processes that occurred below or at that length scale.And,in fact,observational evidence tells us that with respect to its pristine conditions the intra-cluster medium (ICM)gas has been metal enriched by star formation processes,magnetized in some fashion and shock heated.
Perhaps spurred by observational progress,in recent years the subject of non-thermal process in,and their impact on,the ICM has attracted much attention.In this regard,V¨o lk et al.(1996)pointed out that metal enrichment by galactic winds is likely accompanied by inter-galactic termination shocks where cosmic-ray (CR)acceleration might take place.In addition,En?lin et al.(1997)studied the case of CR protons escaping from of radio jets into the ICM and Kronberg et al.(2001)that of black hole magnetic energy injection into the inter-galactic medium (IGM).The issue of CR ions produced at cosmic shocks (Miniati et al.2000)and their possible dynamical role was addressed by Miniati et al.(2001b)by means ?fm@MPA-Garching.MPG.DE
2 F.Miniati
of a cosmological simulation of structure formation which included direct treatment of acceleration,transport and energy losses of a CR component.They found that if shock acceleration takes place with some e?ciency,CR ions,due to their long lifetime against energy losses and to their e?cient con?nement by magnetic irregularities(V¨o lk et al.1996;Berezinsky et al. 1997),may accumulate inside forming structures and store a signi?cant fraction of the total pressure there(Miniati2000; Miniati et al.2001b).
In order to assess the level of non-thermal activity in groups and clusters direct observations of the associated non-thermal emission is required.In particular,the integratedγ-ray photon?ux above100MeV produced in the decay of neutral π-mesons generated in p-p inelastic collisions,allows direct determination of the CR ion content and the pressure support that they provide.For the sources of CRs mentioned in the previous paragraph,the authors that investigated them estimated aγ-ray?ux above100MeV that should be measurable by future targeted observations(V¨o lk et al.1996;Berezinsky et al. 1997;En?lin et al.1997;Blasi1999;Atoyan&V¨o lk2000;Miniati et al.2001b).However,in a recent paper Miniati(2002b) computed the inverse Compton(IC)γ-ray emission from shock accelerated CR electrons scattering o?Cosmic Microwave Background(CMB)photons and for typical clusters of galaxies found it comparable to theγ-ray emission associated to π0-decay.The importance of the former process was recently pointed out by Loeb&Waxman(2000)in the context of the unexplained cosmicγ-ray background(see also Miniati2002b;Keshet et al.2003).According to results of Miniati(2002b),the γ-ray?ux from the leptonic component scales with the group/cluster temperature less strongly than its hadronic counterpart (Miniati et al.2001b).This means that it should dominate theγ-ray emission of small structures.In addition,as we shall see in the ensuing sections,for conventional acceleration parameter valuesγ-ray from IC emission andπ0-decay can be of the same order even for a Coma-like cluster.Thus,the aimed detection ofγ-ray emission from clusters may not necessarily re?ect the CR hadronic component and this should be borne in mind for a correct interpretation of the observational results.
With the objective of a correct diagnostic of futureγ-ray observational results,in this paper we compute the radiation spectrum and spatial distribution of non-thermal emission from groups/clusters of galaxies.We inspect the case of CR ions and electrons accelerated at structure shocks as well as secondary electron-positrons(e±).However,we neglect the CRs that might originate at galactic wind termination shocks(V¨o lk et al.1996)and radio galaxies.The computed spectrum extends from hard X-ray(HXR)at10keV up to extremeγ-ray at10TeV.We examine the following emission processes:IC and non-thermal bremsstrahlung(NTB)from shock accelerated electrons,IC emission from e±andγ-rays fromπ0-decay produced by CR ions.We?nd that NTB emission is negligible throughout the spectrum.IC emission from shock accelerated electrons should dominate at HXR energies and be comparable to the?ux fromπ0-decay atγ-ray energies.Finally,IC emission from e±is typically below these two components at all photon energies.However,we show that all of these three emission components (i.e.,e?,e±andπ0)can potentially be individually measured by futureγ-ray detectors with imaging capability by taking into account their peculiar spatial distribution and spectral properties.In particular we?nd that IC emission from shock accelerated CR electrons dominates the emission at the outer regions of galaxy clusters,provided that at least a fraction of a percent of the shock ram pressure is converted into CR electrons.Therefore,a clear detection of this component and of its spatial distribution will allow us direct probing of cosmic accretion shocks.
The paper is structured according to the following rationale:The numerical models for both the large scale structure and the CR evolution are described in§2.The results on the non-thermal emission are presented in§3,discussed in§4and?nally summarized in§5.
2THE NUMERICAL MODEL
The objective of this study is to investigate the spatial and spectral properties of nonthermalγ-ray emission from CR electrons and ions accelerated at large scale structure shocks surrounding clusters/groups of galaxies.The task at hand requires accurate modeling of:(1)the large scale structure shocks where CR acceleration occurs;(2)the distribution within galaxy clusters of the baryonic gas which provides the CR targets for production ofπ0and secondary e±;and(3)the spatial propagation and energy losses of the CRs.Furthermore,for a consistent description of the CR ions it is important to account for(the acceleration due to)all the shocks that have processed the gas that ends up within the GCs themselves.That is because CR ions with momenta above1GeV/c and up to～1018eV/c have a lifetime against energy losses longer than a Hubble time.
This suggests that the formation of clusters and groups of galaxies be followed ab initio.Furthermore in order to reproduce meaningfully the shocks where the CR acceleration occurs,this should be done within the“natural”framework of cosmological structure formation.For this reason we have carried out a numerical simulations of structure formation which follows the evolution of both the gas and dark matter in the universe starting from fully cosmological initial conditions.Details on the cosmological simulation are given below in§2.1.In addition,with the numerical techniques described in the following sections (§2.2)we were also able to include the evolution of CR ions,electrons and secondary e±by accounting explicitly for the e?ects of CR injection(at shocks and through p-p inelastic collisions),di?usive shock acceleration,energy losses and spatial propagation.Thus in addition to the gas distribution the simulation will also provide us with the information about both the spatial and momentum distribution for each CR components.In particular this information will be used in the§3to compute the spatial and spectral distribution of nonthermalγ-ray emission within gravitationally collapsed structures.Some of these results were already preliminarily presented in Miniati(2002a,c).Also,the simulation upon which the study is based,has been used by the author to study the contribution ofγ-ray emission from CRs in the di?use intergalactic medium to the cosmicγ-ray background(Miniati2002b).
In the following subsections we describe in further detail the salient features of the employed numerical techniques.These
3 are basically the same as those used in previous related studies(Miniati et al.2001b,a;Miniati2002b,c),and the reader familiar with them can skip directly to the result section in§3.
2.1Large Scale Structure
The formation and evolution of the large scale structure is computed by means of an Eulerian,grid based Total-Variation-Diminishing hydro+N-body code(Ryu et al.1993).We adopt a canonical,?atΛCDM cosmological model with a total mass density?m=0.3and a vacuum energy density?Λ=1??m=0.7.We assume a normalized Hubble constant h≡H0/100 km s?1Mpc?1=0.67(Freedman2000)and a baryonic mass density,?b=0.04.The simulation is started at redshift z?60. The initial density?eld is homogeneous except for perturbations generated as a Gaussian random?eld and characterized by a power spectrum with a spectral index n s=1and“cluster-normalization”σ8=0.9.The initial velocity?eld is then computed accordingly through the Zel’dovich approximation.
The choice of the computational box size is a compromise between the need of a cosmological volume large enough to contain a satisfactory sample of collapsed objects and numerical resolution requirements.Ideally one would like to be able to produce rich clusters of galaxies of the size of Coma cluster because they can be compared more easily with observations. However,in order for such massive objects to develop out of the initial conditions described above,the size of the computational box should be at least a few hundred h?1Mpc in size.Although this could be achieved with Adaptive Mesh Re?nement techniques,with the available computational resources and the code employed here the above box size would impose an unacceptably coarse spatial resolution.Therefore the size of the computational box is set to L=50h?1Mpc.Given the relatively small box size the dimensions of the largest collapsed objects,with a core temperature T x?2?4keV,are modest. Therefore,when in§3.4and§3.5we make predictions for a Coma-size cluster(with T x?8keV)the computed simulation results will be rescaled appropriately.The details of the rescaling procedure will be described in those sections.
Finally,the dark matter component is described by2563particles whereas the gas component is evolved on a comoving grid of5123zones.Thus each numerical cell measures about100h?1kpc(comoving)and each dark matter particle corresponds to2×109h?1M⊙.
2.2Cosmic-rays
In addition to the baryonic gas and dark matter,the simulation also follows the evolution of three CR components,namely CR ions and electrons injected at shocks and secondary e±generated in p-p inelastic collisions of the CR ions with the gas nuclei.This is achieved through the code COSMOCR fully described in Miniati(2001,2002b).
The dynamics of each of these CR component consist of the following processes:injection,di?usive shock acceleration, energy losses/gains and spatial propagation.In this section we describe how each of these processes is modeled numerically and how it is implemented in the simulation.We notice from the outset that CRs are treated as passive quantities,meaning that their dynamical role is completely neglected both on the shock structure(test particle limit)and the gas dynamics.
2.2.1Injection and acceleration at shocks
The?rst step in order to compute CR injection and acceleration at shocks,is to detect the shocks themselves.Here,as in previous studies,shocks are identi?ed as converging?ows(?·v<0)experiencing a pressure jump?P/P above a thresh-old corresponding to a Mach number M=1.5.For the identi?ed shocks both mass?ux and Mach number are computed. This is simply done by evaluating the jump conditions experienced by the?ow as reproduced in the numerical simulation (e.g.,Landau&Lifshitz1987).Shocks are found both around?laments and groups/clusters.In the latter case they show a rather complex structure(see§3.1and Miniati et al.2000,for a detailed description of cosmic shocks).An example of gas density distribution,velocity?eld and shock structure within a collapsed object is provided in Fig.1,which illustrates the dominance of asymmetry in the accretion?ows and the existence of strong accretion shocks at the outskirts region followed by weaker ones in the inner regions.A detailed description of Fig.1is delayed until§3.1and we shall now focus on the description of the adopted CR injection/acceleration scheme.
As we shall see in the following,the mass?ux through the shock and the shock Mach number determine two important quantities:respectively the CR injection rate and the shape of the accelerated distribution function.Injection and di?usive acceleration at shocks take place on spatial scales of the order of the particles di?usion length which is
λd(p)=1.1 E0.1μG ?1 u s
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For the injection of CR ions we follow the thermal leakage prescription(Ellison&Eichler1984;Quest1988;Kang&Jones 1995).That is,the post-shock gas is assumed to thermalize to a quasi-Maxwellian distribution function characterized by a temperature T shock.The ions/protons in the high energy tail of such distribution are assumed to be able to leak back upstream and undergo the acceleration mechanism.The momentum threshold for such injection is set to p inj=2c1m p
GeV B102Km s?1 ?1yr(2)
is much shorter than the simulation dynamical time(computational time-step).Thus,as in previous studies,we assume that the injected CRs are accelerated instantly and distribute them in momentum according to a power-law distribution extending from injection up maximum energy,i.e.,
f(p)=f0 p
5 equation,we de?ne a grid in momentum space by dividing it in N p logarithmically equidistant intervals(momentum bins).For each mesh point of the spatial grid,x j,the CR distribution function is then described by the following piecewise power-law (Jones et al.1999;Miniati2001)
f(x j,p)=f j(x j)p?q i(x j),1
?n(x j,p i)
a
?·[u n(x j,p i)]+ ˙a3a?·u p+b?(p) 4πp2f(x j,p i) p i p i?1+Q(x j,p i).(9) Here n(x,p i)= p i p i?14πp2f(x j,p)dp is the comoving number density of CR in the i-th momentum bin and Q(x j,p i)is a comoving source term,i(x j,p),describing either shock injection or e±generated in p-p collision and integrated also over the i-th momentum bin.In addition,a is the cosmological expansion factor,˙a/a=H(z)is the Hubble parameter de?ning the cosmic expansion rate.Thus the?rst part of the second term in eq.(9)describes adiabatic losses/gains.Finally,b?(p) represents energy losses due to Coulomb collision,bremsstrahlung,IC and synchrotron emission for electrons and e±;and Coulomb and p-p inelastic collision for ions.All of them are fully described in Miniati(2001).In eq.(9)u is the velocity?eld of the baryonic gas to which the CRs are assumed to be coupled through a background magnetic?eld.Thus once accelerated the CRs are passively advected with the gas?ow.Notice that eq.(9)does not include the di?usion term present in the ordinary di?usion-convection equation.This is because,as pointed out in Miniati(2001,see also Jones et al.1999)for typical?ows in the simulation with u～a few100km s?1di?usion over spatial scales?x?100kpc is much slower than advection and can be neglected away from shocks.This is true for values of the di?usion coe?cients typically assumed in the literature(see,e.g., V¨o lk et al.1996;Berezinsky et al.1997).The value inferred for these di?usion coe?cients is based on the assumption that the magnetic?eld?uctuations are described by a turbulent spectrum and that the total magnetic?eld energy density is that characteristic ofμG strong magnetic?elds.
Integrating eq.(9)is su?cient for an accurate treatment of the ionic component.In this case,after advancing the solution to eq.(9)one time-step,both f j(x j)and q i(x j)de?ning the CR distribution function in eq.(8)are computed based on the updated values of n(x j,p i)and by assuming that f(x j,p)is continuous at each bin interface.
For CR electrons,however,severe energy losses and distribution cuto?s render their numerical treatment much more delicate.For this reason for this component in addition to n(x,p i)we also follow the correspondent bin kinetic energy ε(x,p i)=4π p i p i?1p2f(x,p i)T(p)dp.Here T(p)=(γ?1)m e c2is the particle kinetic energy andγ=[1?(v/c)2]?1/2is the Lorentz factor.The equation describing the evolution ofε(x j,p i)is obtained in analogy to eq.(9),by integrating over the i-th momentum bin the same di?usion-convection equation(more properly a kinetic equation)that has been multiplied by T(p). In comoving units this reads(Miniati2001)
?ε(x j,p i)
a ?·[uε(x j,p i)]+ b(p)4πp2f(x j,p)T(p) p i p i?1? p i p i?1b(p)4πcp3f(x j,p)
m e c2+p2
dp+S(x j,p i),(10)
where S(x j,p i)=4π p i p i?1i(x j,p i)p2T(p)dp,b(p)now includes both adiabatic loss terms and those described by b?(p),and the third term on the right hand side includes contributions from CR pressure work and sink terms.The evolution of the CR electrons is obtained by integrating numerically eq.(9)and(10)through a semi-implicit scheme described in Miniati(2001, 2002b).The slope of the distribution function at each grid point and momentum bin is then determined self consistently by the values of n(x j,p i)andε(x j,p i).
However,for the simulated CR electrons and e±with momenta between102GeV/c and20TeV cooling time due to IC losses,τcool～105?107yr,is much shorter than the typical computational time-step,?t～107?108yr.Thus,because uτcool?u?t ?x(Courant condition),these particles only exist in grid cells where the source term i(x j,p)=0and will not propagate outside it.Thus,as discussed in(Miniati2002b),in this case it is computationally advantageous and numerically correct to take directly the steady state solution to eq.(9)as
n(x j,p i)=4π p i p i?1p2f c(x j,p i)dp=?4π p i p i?1dp
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Figure1.Color image showing the density distribution on a plane passing through the center of the collapsed object.It is in units of cm?3and the various level correspond to values indicated by the colorbar on the right of each panel.Arrows describe the velocity?eld on the same plane.Their number has been reduced by a factor nine for clarity purposes.Finally,blue contours indicate the isolevel of compression(?·v<0)where shoks occur.Narrow countour features correspond to location of strong shocks.Left and right panel correspond to X-Y and Z-X coordinate planes respectively.Axis scales are measured in h?1Mpc.
20TeV given by eq.(11).Our tests indicate that increasing the number of momentum bins for either CR component(ions and electrons)does not a?ect the results.
3RESULTS
In the following sections we present results based on the simulation just described on the emission of high energy radiation from CRs in groups/clusters of galaxies.In particular,we inspect the spectra,radial dependence and synthetic images for various emission mechanisms and for each computed CR population.For the purpose we excise out of the simulation box regions containing individual virialized objects.For each of them we compute the nonthermal radiation produced through emission processes described below,by using the simulated hydro(density)and CR(distribution functions)quantities.For simplicity in the following we focus only on a couple of simulated virialized objects,although the studied radiation properties are independent of our peculiar choice.Finally,in§3.2and§3.3we describe the qualitative properties of the nonthermal radiation whereas in§3.4and§3.5we will attempt quantitative predictions for a Coma-like cluster of galaxies.
3.1Density and Velocity Structure
First we consider the largest virialized object in the computational box.This is characterized by an X-ray core temperature T x?4keV.The object is partially described in?g.1where the two panels show two dimensional cuts across the coordinate planes X-Y and Z-X respectively of the density distribution(in color scale),the velocity?eld(arrows)and the location of shocks(contours).The gas density is n c～6×10?3cm?3at the core and drops by about2orders of magnitudes at a distance of a few Mpc.The tree-dimensional velocity characterizing the accretion?ows is typically of order of103km s?1. The velocity?eld converges toward the mass concentration and becomes quite complex close to it.As described in more detail in Miniati et al.(2000),this is due to the large asymmetry of accretion?ows.Noticeably,stream along?laments,which carry more momentum,reach closer to the central regions before being shocked.As a result the ensuing shock structure is also complex.The largest shocks are found up to～5h?1Mpc from the core of the collapsed object.Additional shocks are present at intermediate distances,especially along?laments as already pointed out.These shocks have Mach numbers that range from up to～100in the most outlining regions,to3?10along?laments.Close to the high temperature core some discontinuities are also found.However,as it is apparent from the broad features of the isocontours,these are weak,tran-sonic shocks.
For the selected object the energy in CR ions is E CR?6×1061erg in the inner1.5Mpc and corresponds to about26% of the thermal pressure within the same volume.We notice the presence in the neighborhood of the collapsed objects of a small structure which is is marginally visible above and to the right of it,in the left and right panels of?g.1respectively.As we shall see in the following,this will a?ect(particularly,will increase)the computed radiation spectrum at large radii.
3.2Radiation Spectrum
In?g.2the quantity‘εLε’in units‘keV s?1’associated with the selected object is plotted as a function of photon energy,ε.‘Lε’is the volume integrated spectral power and is de?ned as
Lε(ε,R)= V(R)dV jε(ε)(12)
where jε(ε)is the spectral emissivity in units‘keV cm?3s?1keV?1’and V(R)=4πR3/3is an integration volume de?ned by a radius R.Here we set R=5h?1Mpc in order to account for the shock accelerated electrons which are typically located away from the cluster center(see?g1).We consider the following emission processes:π0-decay(solid line),IC emission from shock accelerated electrons(dotted line)and from secondary e±(dashed line)scattering o?CMB photons,and NTB from shock accelerated electrons(dot-dashed line).Thermal X-ray photons,of energyεX,can be neglected as seeds for IC emission ofγ-rays with respect to CMB photons to?rst order approximation.In fact,although the number of available IC scattering electrons is larger by a factor( εX / εCMB )(q?3)/2this is not su?cient to compensate for the lower number of seed photons(especially for?at CR distributions such as those produced at accretion shocks)and for the IC cross-section suppression due to Klein-Nishina e?ects which enter for x?1.7[ε(GeV)εX(keV)]1/2 1(see also,e.g.,En?lin&Biermann 1998;Berrington&Dermer2002).
According to?g.2,both emission fromπ0-decay and IC from primary e?contribute signi?cantly to theγ-ray emission above100MeV.The actual proportion between these two components is determined by two factors:(a)The ratio of accelerated electrons and ions at relativistic energies,which has never been measured for cluster shocks.It is represented by the parameter R e/p and was set to10?2in the current study.(b)The temperature of the group/cluster,from which theγ-ray luminosity
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Figure2.Radiation spectrum extending from10keV up to10TeV produced by the following emission processes:IC(dotted)and NTB (dash-dotted)from shock accelerated CR electrons,IC emission from e±(dashed)andγ-rays fromπ0-decay produced by CR ions(solid). fromπ0-decay and IC emission depend di?erently.In fact,for both processes the temperature dependence is expected to be of power-law type,but with a moderately steeper index for the hadronic emission component(Miniati et al.2001b;Miniati 2002b).Thus the relative strength of these two contributions as presented in?g.2is only indicative at this stage.However,it is within the objectives of the present paper to investigate possible strategies for their experimental determination.At HXR and softγ-ray energies below10MeV,IC from primary e?dominates the spectrum although the contribution from e±is not negligible.However,as we shall see in the next sections,this source of emission should also be detectable at HXR energies owing to its di?erent spatial distribution with respect to the primary e?.
Noticeably the radiation spectra of these three components are rather“?at”.This indicates that the primary CR ions and e?’s were generated in shocks of at least moderate strength,i.e.,M 4.Indeed,for the hadronic component,theγ-ray emissivity from a CR ion distribution f(p)∝p?q p,is jε(ε)∝ε?(4q p?13)/3(cf.Mannheim&Schlickeiser1994).Thus,the case of an emission spectrum such thatεLε∝εjε∝ε0,implies a CR ion distribution with q p?4.As for both primary electrons and secondary e±,because of the severe energy losses su?ered by these particles,the emission spectrum must be related to their steady state distributions.As pointed out at the end of§2.2,this is characterized by a steady-state log-slope, q ss=q source+1,that is steeper by one unit with respect to the source spectrum.The produced IC emissivity is of the form jε(ε)∝ε?(q ss?3)/2which satis?es the conditionεjε∝ε0for q ss～5and,therefore,q source～4.For primary e?this straightforwardly requires that these CRs were accelerated at strong shocks.For secondary e±with a source term of the form
f e±(εe±)∝ε?4(q p?1)/3
e±(cf.Mannheim&Schlickeiser1994),it requires that the parent CR ions were accelerated at strong
shocks and that their log-slope distribution be q p?4.
Finally,NTB is negligible for all purposes.Notice that latter component includes also the contribution from trans-relativistic electrons.These manage to propagate away from the acceleration region(i.e.,a shock)for a short time before being re-absorbed into the thermal pool due to Coulomb losses.This trans-relativistic component enhances the NTB emissivity with respect to that produced by the bulk of the relativistic electrons.In fact,it dominates the emission below10MeV causing
a sort of bump in the shape of the spectrum about this photon energy(dot-dashed line in?g.2).
3.3Luminosity Volume Dependence
We now inspect the spatial distribution of the emissivity associated with the various processes presented in the previous section.As anticipated there,such information allows in principle a measurement of each individual emission component (except NTB).With this in mind,in?g.3we show the radial dependence of the integrated photon luminosity above100keV, that is
Lγ(>100keV,R)= >100keV Lε(ε,R)dε(13)
where R,the radius de?ning the integration volume,is now a varying parameter.Obviously,at these low energies there is no contribution from the CR ionic component.Thus the diagram contains only three curves illustrating IC emission from primary
8 F.Miniati
Figure3.Radial dependence of the integrated photon luminosity above100keV for the following emission processes:IC(dotted)and NTB(dash-dotted)from from shock accelerated CR electrons,IC emission from e±(dashed).
Figure4.Radial dependence of the integrated photon luminosity above100MeV for the following emission processes:IC(dotted)and NTB(dash-dotted)from shock accelerated CR electrons,IC emission from e±(dashed)andπ0-decay(solid).
e?(dotted)and secondary e±(dashed)and,for completeness,NTB from e?(dot-dashed).This plot clearly shows how the radiation emitted by primary e?and secondary e±originates in spatially separate regions.In fact,the latter component saturates quickly within the central Mpc and dominates the total photon production there.On the contrary,the contribution from the former component becomes signi?cant only at a distance of about1.5h?1Mpc from the cluster center and keeps increasing up to several Mpc from there.
The situation is analogous atγ-ray energies.This is illustrated in?g.4where the integrated photon luminosity above 100MeV,Lγ(>100MeV,R)-de?ned by an equation analogous to eq.(13)-is plotted versus radial distance.Here it is
9
Figure5.Synthetic map of the integrated photon?ux above100keV(top)and100MeV(bottom)in units“ph cm?2s?1arcmin?2”from IC emission by secondary e±(top-left),primary e?(top-right,bottom-right),andπ0-decay(bottom-left).Each panel measures15 h?1Mpc on a side.
the emission fromπ0-decay(solid line)that saturates at relatively short distances from the cluster center and dominates the photon emission in the inner regions.As before,IC emission from primary e?(dotted line)reaches a substantial level only outside a distance～1.5h?1Mpc.IC emission from e±(dashed line)is now unimportant and,as in the previous case,NTB from primary e?(dot–dashed line)is completely negligible.
The radial dependence of the various radiation components presented in?g.3and4re?ects both the spatial distribution of the emitting particles as well as the nature of the emission process.Thus,on the one hand,both e±andπ0are produced at the highest rate in the densest regions where both the parent CR ions and target ICM nuclei are most numerous.Consequently, e±IC andπ0-decay emissivities are strongest in the cluster inner regions and quickly fade toward its outskirts.Notice that in the case of?g.3and4this behavior is slightly altered by the presence of a nearby object(similar to the case in?g.5below) which causes these integrated photon luminosities to slowly grow even after few core radii instead of leveling out.
On the other hand,because of severe energy losses,the emitting high energy primary e?are only found in the vicinity of strong shocks where they are being accelerated.Notice that in this case the shocks must be strong so that enough particles are accelerated to high energies to produce appreciable emission.Thus,because the strongest shocks are located at the cluster outskirts rather than at its core where the ICM temperature is already high(?g.1;also cf.Miniati et al.2000),the spatial distribution of the emissivity is now reversed with respect to the previous case.Notice that,in line with this analysis,the drop of IC emissivity(from primary e?)toward the inner regions is slightly more abrupt atγ-ray energies(?g.4)than at HXR energies(?g.3).
3.4Synthetic Maps
In this section we present synthetic maps produced by non-thermal emission processes.These maps were obtained upon simple integration along the line of sight of the photon emissivity in the thin plasma approximation.We use them to study
10 F.Miniati
Figure6.Left,Right:Synthetic map of the total photon?ux above100keV(left)and100MeV(right)in units‘ph cm?2s?1arcmin?2’. Left panel includes IC emission from primary e?,secondary e±and NTB.Right panel includes emission due toπ0-decay,IC from primary e?,secondary e±and NTB(c).Center:Synthetic map of the bolometric X-ray emission from thermal bremsstrahlung in units ‘erg cm?2s?1arcmin?2’.Each panel measures15h?1Mpc on a side.
the morphology of the emitting region.In addition we investigate which emission component contributes most to the radiation spectrum when the latter is extracted from di?erent spatial regions.Unlike plots in?g.3and4,using synthetic maps will automatically account for projection e?ects.
We will attempt to relate the present analysis to the case of Coma cluster which,for its large size and relative vicinity, is likely the best source candidate for the detection of non-thermal,high energy radiation.Our calculations are now based on a di?erent collapsed object extracted as before from the simulation box but relatively more isolated(and therefore better suited for the current purpose)than the previous one.It has a temperature of a few keV,a core density of2×10?4cm?3.
11 This object is smaller than Coma cluster and for that reason we will renormalize the emission components as follows.1As for the hydro quantities we retain the temperature and density pro?les produced in the simulation but renormalize them by (multiplying each of them by)the ratio of the observed and simulated core values.The following observed values for Coma cluster are taken:a temperature T x=8.2keV(Arnaud et al.2001)and a gas density of n gas?3×10?3cm?3.The thermal energy within1Mpc computed according to temperature and density thus renormalized corresponds to1.6×1063erg.
For the IC emission from primary e?we have used the recent?nding of Miniati(2002b)indicating that theγ-ray?ux due to this process scales with the X-ray temperature as Fγ∝T2.6
x
.The total number of secondary e±,N e±,and the associated
IC?ux F±
IC ,are normalized by assuming that these particles are responsible for producing Coma radio halo through emission
of synchrotron radiation.For a given measured radio?ux,S sy,and an assumed average ICM magnetic?eld strength, B , the total number of emitting particles scales as
F±IC∝N e±∝
S sy
1+U B/U CMB
(15) based on this and eq.(14)one?nds
Fπ0→γγ∝N cri ρgas ∝S sy 1+U B
B 1+α
,(16)
where U B= B 2/8πand U CMB are the energy density in magnetic?eld and CMB radiation?eld,respectively.For the synchrotron?ux needed in(14)and(16)we adopted the value S1.4GHz=640mJy measured at1.4GHz by Deiss et al.(1997). When extracting synthetic spectra in the next section we consider two cases for the magnetic?eld strength corresponding to B ～0.15μG and B ～0.5μG(Kim et al.1990).The former choice implies a ratio of CR ions to thermal energy about 30%as opposed to almost4%for the latter.This second case is assumed to produce the synthetic maps described below. Notice that its“renormalized”values would imply a lower CR ion acceleration e?ciency then described in§2.2.1and would also correspond to a parameter R e/p～0.05.
It is worth pointing out that according to eq.(14)and(16)smaller values of B than assumed here would imply corre-spondingly higher?uxes of IC emission andγ-ray fromπ0-decay.For very weak?elds,then,most of the assumed radio emitting particles could not be of hadronic origin due to EGRET upper limits onγ-ray?ux from Coma cluster(Blasi&Colafrancesco 1999;Miniati et al.2001a,cf.also?g.8).In the opposite limit of stronger magnetic?elds than assumed here,less emitting particles would be required to produce the same radio emission.As a consequence,the associated IC?ux is reduced propor-tionally to the inverse of the magnetic energy density[cf.eq.(14)].However,as indicated by the expression in eq.(16),for α?1and B ?
12 F.
Miniati
Figure 7.Synthetic spectra extracted from the core region (top)and the outskirts region (bottom).
3.5Synthetic Spectra
In order to assess the separability of the di?erent emission components,in ?g.7we have produced synthetic spectra taken from a core (top)and an outskirts region (bottom).The extension of the core region corresponds to an angular size of 1o (or 2Mpc diameter at the red-shift of Coma cluster)whereas the outskirts region is de?ned as an annular ring with inner and outer radii of 0.5o and 1.5o (or 1Mpc and 3Mpc at the red-shift of Coma cluster)respectively.For the core region we consider two values for the magnetic ?eld strength (0.15and 0.5μG).For each emission region in ?g.7we plot the predicted ?ux contribution due to the main emission processes considered so far namely,IC emission from primary e ?(dot)and secondary e ±(dash and dot–dash),and π0-decay (solid and long–dash).Notice that as the magnetic ?eld strength is increased from 0.15to 0.5μG the ?uxes due to IC from e ±and π0-decay drop by about a factor ～10(cf.eq.14and 16).Fig.7shows that at high photon energy (>100MeV)the spectrum of radiation is indeed dominated by π0-decay in the core region (top panel)and by IC emission from primary e ?in the outer region (bottom).In the latter case,the residual π0-decay component is actually due to the presence in the ?eld of view of a small structure north-west of the selected object (see ?g.5;since this is not part of the main cluster,we do not renormalize it according to various magnetic ?eld values as it was done for the core emission).In principle any contribution such as this can be removed by excision of emission regions associated with thermal X-rays from structures appearing in the ?eld of view.Below ～100MeV the ?ux from the outskirts region is still strongly dominated by IC emission from primary e ?.The contribution from the latter is much reduced in the narrower ?eld of view of the core region.However,for the larger ?eld case it is still signi?cant and at the level of IC emission from secondary e ±.The relative amount of radiation ?ux from these two components further depends on the actual shock structure subtended by the ?eld of view.Nevertheless the point is that for magnetic ?eld of a few tenths of μG they are expected of comparable intensity.In order to separate them out one could measure the radiation ?ux as a function of radial distance in the outer regions,where it presumably is only due to CR electrons accelerated at accretion shocks,and then extrapolate its value for the core region.For this purpose,an angular resolution ～10′or so should be su?cient.
Finally,above the spectra are characterized by a cuto?which appears at photon energies about 1TeV.For IC emission from primary e ?,for which this feature is sharpest,this is mostly due to the maximum momentum,p max =20TeV,of the accelerated particles.For the other processes,it is rather caused by absorption due to the reaction γγ?→e +e ?.In fact,pair production becomes important at TeV energies due to the presence of di?use background radiation ?eld at optical/infrared wavelengths.The spectra in ?g.7were thus corrected through a factor exp(?τγγ)and by assuming an optical depth for pair production (e.g.,Coppi &Aharonian 1999)
τγγ(εγ)?0.14 εγ
2×10?3eV cm ?3 z
Coma
13
Figure8.Total radiation spectra extracted from the same spatial regions illustrated in?g.7–core for two di?erent magnetic?eld values(dot and long–dash),outskirts(short dash)and their summation(solid and dot–dash)–compared to nominal sensitivity limits of futureγ-ray observatories(thick-solid lines).For INTEGRAL-IBIS imagers(ISGRI and PICsIT)the curves correspond to a detection signi?cance of3σwith an observing time of106s.All other sensitivity plots refer to a5σsigni?cance.EGRET and GLAST sensitivities are shown for one year of all sky survey whereas for Cherenkov telescopes(MAGIC,VERITAS and HESS)for50hour exposure on a single source.
4DISCUSSION
In?g.8we plot sensitivity limits of plannedγ-ray observatories together with the total radiation spectra from the core(dotted line)and outskirts(dashed line)regions that were presented in?g.7.The sensitivity limits are plotted for point sources.For extended sources it will be worse by roughly a factor that goes asθsource/θinst that is the ratio between the source size and the angular resolution of the instrument.For Cherenkov telescopesθinst～0.1o In any case,according to this plot,several future experiments should be sensitive enough to detect the computed non-thermal emission at most photon energies.In particular the IBIS imager onboard INTEGRAL should readily measure the?ux between100keV and several MeV.In addition GLAST and Cherenkov telescopes should be able to detect both the coreγ-ray emission fromπ0-decay as well as the IC?ux directly produced at accretion shocks above100MeV and10GeV respectively.One should notice that resolving sharp and complex structures such as those reported in?g.5and6is a real challenge,especially for coded mask techniques employed by several current and plannedγ-ray imagers.Although a nominal resolution of～10′in principle should be su?cient to identify some of the bright shock features,the?nal outcome of such measurements will hinge on the actual source?uxes and instruments performance.
The measurement of the non-thermal radiation spectra at several photon energies spanning the range illustrated in?g. 7provides important information about the physical conditions in clusters.First,besides the mere detection of HXR and γ-ray radiation from clusters,important information is embedded in the spatial distribution of their surface brightness.In particular the extended emission component corresponds to the location of accretion shocks.Merger shocks have occasionally been observed in the core of clusters as relatively weak temperature jumps.However,strong accretion shocks have yet to be observed directly.Thus,detection and imaging of IC emission from primary electrons would provide an opportunity to directly observe these accretion shocks given their wide angular extension and their unique morphology.
In addition,the?ux about100keV will give us the?rst direct estimate of the energy density of CR electrons and,together with radio measurements,will allow an estimate of(or upper limits on)the ICM magnetic?eld.In this respect for a correct interpretation of the data it will be necessary to account properly for the?uxes contributed by both the CR electrons in the core and those directly accelerated at the outlining shocks.In fact,the former likely propagate in relatively highly magnetized regions and produce a substantial radio emission.On the other hand,because of the expected decline of the magnetic?eld strength toward the outer regions,the latter might generate only a weak radio emission but,nonetheless,a strong IC?ux. Since,as shown in?g.7,the HXR?ux produced by this second component can easily dominate the total?ux at this spectral range(the details will depend on the assumed normalizations on acceleration e?ciency and average magnetic?eld strength) separating the contributions from CRs in the cluster core and external accretion shocks will be an important step for correctly measuring ICM magnetic?elds.A similar point was qualitatively discussed already in Miniati et al.(2001a)and was also
14 F.Miniati
addressed in Brunetti et al.(2001)in the context of a their“multiphase”acceleration model.In particular Brunetti et al. (2001)pointed out that the magnetic?eld in the outer region,where most of the IC emission is produced due to the larger emitting volume,can be much lower than in the core where the radio emission originates.In that model all the radio and HXR emitting particles are accelerated with the same mechanism,whereas in our model the CR electrons that generate the HXR?ux are accelerated in the outer accretion shocks and never make it to the cluster core.
It is worth mentioning that our estimated non-thermal?ux from IC emission above20keV is quite smaller than the recently reported measurements of excess of HXR radiation with respect to thermal emission.For the case of Coma cluster our predictions,as illustrated in?g.8,fall short by a factor of several(Fusco-Femiano et al.1999;Rephaeli et al.1999). An analogous estimate,obtained for A2256after appropriate rescaling,indicates a similar discrepancy by a factor～30 (Fusco-Femiano et al.2000),although the upper limits on A3667are respected by our predictions(Fusco-Femiano et al. 2001).The above discrepancies between prediction and reported detections could be readily improved by assuming that the electron acceleration e?ciency at shocks is larger by an order of magnitude with respect to what assumed here(see also Loeb&Waxman2000).Interestingly,since these electrons are located at the cluster outskirts and need not be responsible for the radio halo emission,their hard X-ray emission would not constraint the ICM magnetic?eld detected through Faraday rotation measures.However,as it is apparent from?g.8,this is at odds with EGRET experimental upper limits on theγ-ray ?ux above100MeV(Sreekumar et al.1996;Reimer1999;Reimer et al.2003),which allow for the acceleration e?ciency adopted here to be increased by at most a factor～2(Miniati2002b).This constraint,however,holds true only as long as:(1)CR e?are accelerated above momenta～100GeV/c and(2)the spectrum of the accelerated particles is not much steeper than computed here.These conditions,though,seem to be easily ful?lled for the case of accretion shocks.Thus,in this respect,if all of the HXR excess emission is di?use in nature and not associated with individual sources,turbulent and/ or second order Fermi acceleration models may provide a more natural explanation because a high momentum cuto?in this range arises more naturally in those models(provided of course enough acceleration e?ciency).
One of the most awaited experiments is related to the measurement of theγ-ray?ux at and above100MeV.This is important in order to convalidate or rule out secondary e±models for radio halos in galaxy clusters(Dennison1980;Vestrand 1982;Dolag&En?lin2000;Blasi&Colafrancesco1999;Miniati et al.2001a)and in order to estimate the non-thermal CR pressure there(Miniati et al.2001b).In this perspective the above authors estimated for nearby clusters theγ-ray?ux produced from the decay of neutral pions produced in p-p inelastic collisions.As pointed out in Miniati(2002b),however,γ-ray from IC emission can also be substantial and at the same level as that fromπ0-decay.Therefore,once again for a correct interpretation of the measurements,separation of these two components will be required.Estimating theγ-ray?ux from IC emission due to shock accelerated CR electrons will be instrumental for addressing another issue of great interest. That is,the contribution of this emission mechanism to the cosmicγ-ray background(Loeb&Waxman2000;Miniati2002b; Keshet et al.2003),for which so far the only constraint is provided by an upper limit based on EGRET experiments(Miniati 2002b).
Photon energies of order～10?100GeV to1?10TeV,will be investigated by Cherenkov telescopes.Such measurements will be complementary to that carried out at lower energies.Because the radiated energy spectrum is directly connected to the distribution function of the emitting particles,measuring the?ux at di?erent energies will provide information about the acceleration mechanism.In particular,the observed spectrum could di?er from our predictions if the accretion shocks were modi?ed by CR pressure.That in fact would cause the radiation spectrum to soften at low energies and to become harder toward higher energies,up to an energy cut-o?(e.g.,Malkov&Drury2001;Berezhko&Ellison1999;Kang&Jones2002). Given the di?erent environmental conditions with respect to supernova remnants,these are interesting issues for a broad exploration of shock acceleration physics.Independent of the shock dynamics,measurements in this photon energy range should help determining the maximum momentum of the accelerated CR electrons and,perhaps,even of CR ions in case these maxima produce spectral cuto?before attenuation byγγabsorption becomes important.For the electronic component this is not excluded and could allow a clearer speci?cation of the energy range in which they can contribute to the cosmic γ-ray background.
5SUMMARY&CONCLUSIONS
We have explored the spatial and spectral properties of non-thermal emission atγ-ray energies.For the purpose we carried out a simulation of structure formation including the evolution of baryonic gas,dark matter,CR ions and electrons accelerated at cosmic shocks as well as secondary e±generated in inelastic p-p collisions.We estimated the radiation?ux between10keV and10TeV from CRs in collapsed structures due to IC emission,π0-decay and NTB and made speci?c predictions for the case of Coma cluster.We have assessed the importance of distinguishing among the contribution from di?erent CR components for a correct interpretation of future experimental results and we have outlined a strategy to do so.Our conclusions are summarized as follows:
?Two main regions for production of non-thermal radiation in clusters/groups of galaxies were identi?ed:the core also bright in thermal X-ray and the outskirts region where accretion shocks occur.
?The chief radiation mechanism at allγ-ray energies in the outskirts region is IC emission from shock accelerated CR electrons,provided that a fraction of a percent of the shock ram pressure is converted into CR electrons.A clear detection of this component and of its spatial distribution will allow us direct probing of cosmic accretion shocks.Such evidence would corroborate recent?ndings about extra-galactic radio emission from large scale shocks(Bagchi et al.2002).
15?In the cluster core,γ-ray emission above100MeV is dominated byπ0-decay mechanism.At lower energies,IC emission from secondary e±takes over.However,IC emission from shock accelerated electrons projected onto the cluster core will not be negligible in general.
?Separating the aforementioned emission components is important for a correct interpretation of the experimental data. This can be achieved in principle by measuring the spatial distribution of the detected emission.
?Measuring the non-thermal spectrum will provide us with knowledge regarding:the injection e?ciency as well as the ram pressure to CR pressure conversion e?ciency for both electrons and ions;the energy range in which IC emission from CR electrons accelerated at accretion shocks can contribute to the CGB;the CR content in galaxy clusters;the dynamical conditions(CR modi?ed or not)of the accretion shocks.
ACKNOWLEDGMENTS
This work was carried out at the Max-Planck-Institut f¨u r Astrophysik under the auspices of the European Commission for the‘Physics of the Intergalactic Medium’.I am grateful to E.Churazov,T.En?lin,M.Gilfanov,F.Aharonian and I.Susumu for useful discussion.The computational work was carried out at the the Rechenzentrum in Garching operated by the Institut f¨u r Plasma Physics and the Max-Planck Gesellschaft.
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This figure "fig1.gif" is available in "gif" format from: https://www.doczj.com/doc/069920567.html,/ps/astro-ph/0303593v1
EVA计算方法 说明: 经济增加值(EVA)＝税后净营业利润（NOPAT）－资本成本（cost of capital） 资本成本＝资本×资本成本率 由上知，计算EVA可以分做四个大步骤：（1）税后净营业利润（NOPAT）的计算; (2)资本的计算；(3)资本成本率的计算；（4）EVA的计算。下面列出EVA的计算步骤，并以深万科（0002）为例说明EVA（2000年）的计算。 深万科（0002）简介： 公司名称：万科企业股份有限公司公司简称：深万科Ａ上市日期：1991-01-29 上市地点：上海证券交易所行业：房地产业股本结构：A 股398711877股，B股121755136 股，国有股、境内法人股共110504928股，股权合计数：630971941股。 一、税后净营业利润（NOPAT）的计算 1．以表格列出的计算步骤 下表中，最左边一列（以IS开头）代表损益表中的利润计算步骤，最右边一列（以NOPAT开头）代表计算EVA所用的税后净营业利润（NOPAT）的计算步骤。空格代表在计算相应指标（如NOPAT）的步骤中不包含该行所对应的项。
损益表中的利润计算步骤 税后净营业 利润 （NOPAT） 的计算步骤主营业务收入 - 销售折扣和折让- - 主营业务税金及附加- - 主营业务成本- 主营业务利润 + 其它业务利润+ 当年计提或冲销的坏帐准备+ - 当年计提的存货跌价准备 - 管理费用- - 销售费用- = 营业利润/调整后的营业利润 + 投资收益+
= 总利润/税前营业利润 - EVA税收调整* - = 净利润/税后净营业利润 2. 计算公式：(蓝色斜体代表有原始数据，紫色下划线代表此数据需由原始数据推算出) （1）税后净营业利润=主营业务利润＋其他业务利润＋当年计提或冲销的坏帐准备—管理费用—销售费用＋长期应付款，其他长期负债和住房公积金所隐含的利息＋投资收益—EVA税收调整 注：之所以要加上长期应付款，其他长期负债和住房公积金所隐含的利息是因为sternstewart公司在计算长期负债的利息支出时，所用的长期负债中包含了其实不用付利息的长期应付款，其他长期负债和住房公积金。即，高估了长期负债的利息支出，所以需加回。 （2）主营业务利润＝主营业务收入—销售折扣和折让—营业税金及附加—主营业务成本 注: 主营业务利润已在sternstewart公司所提供的原始财务数据中直接给出 (3)EVA税收调整=利润表上的所得税＋税率×（财务费用＋长期应
幻灯片1 正态分布 第二章 第七节 一、标准正态分布的密度函数 二、标准正态分布的概率计算 三、一般正态分布的密度函数 四、正态分布的概率计算幻灯片2 正态分布的重要性正态分布是概率论中最重要的分布, 这能够由 以下情形加以说明: ⑴ 正态分布是自然界及工程技术中最常见的分布之一, 大量的随机现象都是服从或近似服从正态分布的．能够证明, 如果一个随机指标受到诸多因素的影响, 但其中任何一个因素都不起决定性作用, 则该随机指标一定服从或近似服从正态分布． 这些性质是其它 ⑵ 正态分布有许多良好的性质, 许多分布所不具备的． ⑶ 正态分布能够作为许多分布的近似分布．幻灯片3 －标准正态分布下面我们介绍一种最重要的正态分布 一、标准正态分布的密度函数若连续型随机变量X 的密度函数为定义 则称X 服从标准正态分布,
记为标准正态分布是一种特别重要的它的密度函数经常被使用, 分布。 幻灯片4 密度函数的验证 则有 ( 2) 根据反常积分的运算有能够推出 幻灯片5 标准正态分布的密度函数的性质若随机变量 , X 的密度函数为 则密度函数的性质为: 的图像称为标准正态( 高斯) 曲线幻灯片6 随机变量 由于 由图像可知, 阴影面积为概率值。对同一长度的区间 , 若这区间越靠近 其对应的曲边梯形面积越大。标准正态分布的分布规律时”中间多, 两头少” . 幻灯片7 二、标准正态分布的概率计算 1、分布函数分布函数为幻灯片8 2、标准正态分布表书末附有标准正态分布函数数值表, 有了它, 能够解决标准正态分布的概率计算.表中给的是x > 0时，①(x)的值. 幻灯片9 如果由公式得令则幻灯片10
怎样理解分布函数 概率论中一个非常重要的函数就是分布函数,知道了随机变量的 分布函数,就知道了它的概率分布,也就可以计算概率了。 一、理解好分布函数的定义: F(x)=P(X≤x), 所以分布函数在任意一点x的值,表示随机变量落在x点左边(X≤x)的概率。它的定义域是(-∞,+∞),值域是[0,1]. 二、掌握好分布函数的性质: (1)0≤F(x)≤1; (2)F(+∞)=1,F(-∞)=0; 可以利用这条性质确定分布函数中的参数,例如: 设随机变量X的分布函数为:F(x)=A+Barctanx,求常数A与B. 就应利用本性质计算出A=1/2,B=1/π. (3)单调不减; (4)右连续性。 三、会利用分布函数求概率 在利用分布函数求概率时,以下公式经常利用。
(1)P(a 经济增加值(eva)计算方式 (四)（Economic value added (EVA) calculation (four)） Next, the calculation method of economic value added is introduced The calculation model of EVA is given below. Computational model of 1 and EVA Economic value added = net operating profit after tax - cost of capital = net operating profit after tax - total capital * weighted average cost of capital Among them: Net operating profit after tax net profit after tax interest expense + = + + minority income this year amortization of goodwill + deferred tax credit balances increase reserve balances increased + + other capitalized research and development costs, capitalized research and development costs in the years of amortization Total capital = common equity + minority interests + deferred tax credit (debit balance is negative) + + (cumulative amortization of goodwill reserve inventory impairment provision for bad debts, etc.) + + + capitalization amount of short-term loans for research and development costs of long term loan + short-term long-term loans due in part EVA 计算方法 说明: 经济增加值(EV A)＝税后净营业利润（NOPA T ）－资本成本（cost of capital ） 资本成本＝资本×资本成本率 由上知，计算EV A 可以分做四个大步骤： （1）税后净营业利润（NOPA T ）的计算; (2)资本的 计算； (3)资本成本率的计算； （4）EV A 的计算。下面列出EV A 的计算步骤，并以深万科（0002）为例说明EV A （2000年）的计算。 深万科（0002）简介： 公司名称：万科企业股份有限公司 公司简称：深万科Ａ 上市日期：1991-01-29 上市地点：上海证券交易所 行业：房地产业 股本结构：A 股398711877股，B 股121755136 股，国有股、境内法人股共110504928股，股权合计数：630971941股。 一、税后净营业利润（NOPA T ）的计算 1． 以表格列出的计算步骤 下表中，最左边一列（以IS 开头）代表损益表中的利润计算步骤，最右边一列（以NOPA T 开头）代表计算EV A 所用的税后净营业利润（NOPA T ）的计算步骤。空格代表在计算相应指标（如NOPA T ）的步骤中不包含该行所对应的项。 损益表中的利润计算步骤 税后净营业 利润 （NOPA T ）的计算步骤 主营业务收入 - 销售折扣和折让 - - 主营业务税金及附加 - - 主营业务成本 - 主营业务利润 - 管理费用 - - 销售费用 - = 营业利润/调整后的营业利润 + 投资收益 + = 总利润/税前营业利润 - EVA 税收调整* - = 净利润/税后净营业利润 2.计算公式：(蓝色斜体代表有原始数据，紫色下划线代表此数据需由原始数据推算出) （1）税后净营业利润=主营业务利润＋其他业务利润＋当年计提或冲销的坏帐准备—管理费用—销售费用＋长期应付款，其他长期负债和住房公积金所隐含的利息＋投资收益—EV A 税收调整 注：之所以要加上长期应付款，其他长期负债和住房公积金所隐含的利息是因为sternstewart公司在计算长期负债的利息支出时，所用的长期负债中包含了其实不用付利息的长期应付款，其他长期负债和住房公积金。即，高估了长期负债的利息支出，所以需加回。 （2）主营业务利润＝主营业务收入—销售折扣和折让—营业税金及附加—主营业务成本注: 主营业务利润已在sternstewart公司所提供的原始财务数据中直接给出 (3)EV A税收调整=利润表上的所得税＋税率×（财务费用＋长期应付款，其他长期负债 和住房公积金所隐含的利息＋营业外支出－营业外收入－补贴收入） (4)长期应付款，其他长期负债和住房公积金所隐含的利息＝长期应付款，其他长期负债 和住房公积金×3～5 年中长期银行贷款基准利率 长期应付款，其他长期负债和住房公积金=长期负债合计—长期借款—长期债券 税率=0.33（从1998年，1999年和2000年） 说明：上面计算公式所用数据大多直接可以在sternstewart公司所提供的原始财务数据中找到（主营业务利润已直接给出）。而长期应付款，其他长期负债和住房公积金所隐含的利息需由原始财务数据推算得出。 3. 计算深万科的税后净营业利润（NOPAT 2000年） 首先计算出需由其他原始财务数据推算的间接数据项－长期应付款，其他长期负债和住房公积金所隐含的利息和EV A税收调整，然后利用计算结果及其他数据计算出NOPA T. （1）长期应付款，其他长期负债和住房公积金所隐含的利息的计算； 单位：元 长期负债合计123895991.54 减：长期借款80000000.00 减：长期债券 ――――――――――――――――――――――――――――― 长期应付款，其他长期负债和住房公积金43895991.54 乘：3～5 年中长期银行贷款基准利率 6.03％ 长期应付款，其他长期负债2646928.29 和住房公积金所隐含的利息 （2）EV A税收调整的计算； 财务费用1403648.37 加：长期应付款，其他长期负债2646928.29 和住房公积金所隐含的利息 加：营业外支出6595016.31 减：营业外收入23850214.53 EV A计算方法 说明: 经济增加值(EV A)＝税后净营业利润（NOPAT）－资本成本（cost of capital） 资本成本＝资本×资本成本率 由上知，计算EV A可以分做四个大步骤： （1）税后净营业利润（NOPAT）的计算; （2）资本的计算； （3）资本成本率的计算； （4）EV A的计算。 下面列出EV A的计算步骤，并以深万科（0002）为例说明EV A（2000年）的计算。 深万科（0002）简介： 公司名称：万科企业股份有限公司公司简称：深万科Ａ上市日期：1991-01-29 上市地点：上海证券交易所行业：房地产业股本结构：A股398711877 股，B股121755136 股，国有股、境内法人股共110504928股，股权合计数：630971941股。一、税后净营业利润（NOPAT）的计算 1．以表格列出的计算步骤 下表中，最左边一列（以IS开头）代表损益表中的利润计算步骤，最右边一列（以NOPA T 开头）代表计算EV A所用的税后净营业利润（NOPA T）的计算步骤。空格代表在计算相 应指标（如NOPA T）的步骤中不包含该行所对应的项。 损益表中的利润计算步骤 税后净营业 利润 （NOPAT） 的计算步骤主营业务收入 - 销售折扣和折让- - 主营业务税金及附加- - 主营业务成本- 主营业务利润 - 管理费用- = 营业利润/调整后的营业利润 + 投资收益+ = 总利润/税前营业利润 = 净利润/税后净营业利润 2.计算公式：(蓝色斜体代表有原始数据，紫色下划线代表此数据需由原始数据推算出) （1）税后净营业利润=主营业务利润＋其他业务利润＋当年计提或冲销的坏帐准备—管理费用—销售费用＋长期应付款，其他长期负债和住房公积金所隐含的利息＋投资收益—EV A 税收调整 注：之所以要加上长期应付款，其他长期负债和住房公积金所隐含的利息是因为sternstewart公司在计算长期负债的利息支出时，所用的长期负债中包含了其实不用付利息的长期应付款，其他长期负债和住房公积金。即，高估了长期负债的利息支出，所以需加回。 （2）主营业务利润＝主营业务收入—销售折扣和折让—营业税金及附加—主营业务成本注: 主营业务利润已在sternstewart公司所提供的原始财务数据中直接给出 (3)EVA税收调整=利润表上的所得税＋税率×（财务费用＋长期应付款，其他长期负债 和住房公积金所隐含的利息＋营业外支出－营业外收入－补贴收入） (4)长期应付款，其他长期负债和住房公积金所隐含的利息＝长期应付款，其他长期负债 和住房公积金×3～5 年中长期银行贷款基准利率 长期应付款，其他长期负债和住房公积金=长期负债合计—长期借款—长期债券 税率=0.33（从1998年，1999年和2000年） 说明：上面计算公式所用数据大多直接可以在sternstewart公司所提供的原始财务数据中找到（主营业务利润已直接给出）。而长期应付款，其他长期负债和住房公积金所隐含的利息需由原始财务数据推算得出。 3. 计算深万科的税后净营业利润（NOPAT 2000年） 首先计算出需由其他原始财务数据推算的间接数据项－长期应付款，其他长期负债和住房公积金所隐含的利息和EV A税收调整，然后利用计算结果及其他数据计算出NOPA T. （1）长期应付款，其他长期负债和住房公积金所隐含的利息的计算； 单位：元 长期负债合计123895991.54 减：长期借款80000000.00 减：长期债券 ――――――――――――――――――――――――――――― 长期应付款，其他长期负债和住房公积金43895991.54 乘：3～5 年中长期银行贷款基准利率 6.03％ 长期应付款，其他长期负债2646928.29 和住房公积金所隐含的利息 （2）EV A税收调整的计算； 财务费用1403648.37 正态分布 第二章 第七节 一、标准正态分布的密度函数 二、标准正态分布的概率计算 三、一般正态分布的密度函数 四、正态分布的概率计算 幻灯片2 正态分布的重要性正态分布是概率论中最重要的分布， 这可以由 以下情形加以说明： ⑴正态分布是自然界及工程技术中最常见的分布 之一， 大量的随机现象都是服从或近似服从正态分布的． 可以证明， 如果一个随机指标受到诸多因素的影响, 但其中任何一个因素都不起决定性作用， 则该随机指标 一定服从或近似服从正态分布． 这些性质是其它 ⑵正态分布有许多良好的性质， 许多分布所不具备的． ⑶正态分布可以作为许多分布的近似分布． 幻灯片3 －标准正态分布 下面我们介绍一种最重要的正态分布 一、标准正态分布的密度函数 若连续型随机变量X的密度函数为 定义 则称X服从标准正态分布， 记为 标准正态分布是一种特别重要的 它的密度函数经常被使用， 分布。 幻灯片4 密度函数的验证 则有 （2）根据反常积分的运算有 可以推出 幻灯片5 标准正态分布的密度函数的性质 ，X的密度函数为 则密度函数的性质为： 的图像称为标准正态（高斯）曲线。 幻灯片6 随机变量 由于 由图像可知，阴影面积为概率值。 对同一长度的区间 ，若这区间越靠近 其对应的曲边梯形面积越大。 标准正态分布的分布规律时“中间多，两头少”. 幻灯片7 二、标准正态分布的概率计算 1、分布函数 分布函数为 幻灯片8 2、标准正态分布表 书末附有标准正态分布函数数值表，有了它，可以解决标准正态分布的概率计算. 表中给的是x > 0时, Φ(x)的值. 幻灯片9 如果 由公式得 令 则 幻灯片10 例1 解 幻灯片11 由标准正态分布的查表计算可以求得， 当X～N(0,1)时， 这说明，X 的取值几乎全部集中在[-3,3]区间内，超出这个范围的可能性仅占不到0.3%. 幻灯片12 三、一般正态分布的密度函数 如果连续型随机变量X的密度函数为 （其中 为参数） 的正态分布，记为 则随机变量X服从参数为 所确定的曲线叫 作正态(高斯）曲线. 幻灯片13 目录 1. 均匀分布 (1) 2. 正态分布（高斯分布） (2) 3. 指数分布 (2) 4. Beta分布（：分布） (2) 5. Gamm 分布 (3) 6. 倒Gamm分布 (4) 7. 威布尔分布（Weibull分布、韦伯分布、韦布尔分布） (5) 8. Pareto 分布 (6) 9. Cauchy分布（柯西分布、柯西-洛伦兹分布） (7) 2 10. 分布（卡方分布） (7) 8 11. t分布................................................ 9 12. F分布 ............................................... 10 13. 二项分布............................................ 10 14. 泊松分布（Poisson 分布）............................. 11 15. 对数正态分布........................................ 1. 均匀分布 均匀分布X ~U（a,b）是无信息的，可作为无信息变量的先验分布。 2. 正态分布（高斯分布） 当影响一个变量的因素众多，且影响微弱、都不占据主导地位时，这个变量 很可能服从正态分布，记作 X~N （」f 2）。正态分布为方差已知的正态分布 N （*2）的参数」的共轭先验分布。 1 空 f （x ）： —— e 2- J2 兀 o' E(X)， Var(X) _ c 2 3. 指数分布 指数分布X ~Exp （ ）是指要等到一个随机事件发生，需要经历多久时间。其 中，.0为尺度参数。指数分布的无记忆性： Plx s t|X = P{X t}。 f （X ）二 y o i E(X) 一 4. Beta 分布（一：分布） f （X ）二 E(X) Var(X)= (b-a)2 12 Var(X)二 1 ~2 EVA 计算方法 说明: 经济增加值（EVA ）=税后净营业利润（NOPAT ）—资本成 本（cost of capital ） 资本成本=资本x 资本成本率 由上知，计算EVA 可以分做四个大步骤： （1 ）税后净 营 业利润（NOPAT ）的计算;（2）资本的计算；（3）资本成本率的计算； （4） EVA 的计算。下面列出EVA 的计算步骤，并以深万科（0002 ） 为例说明EVA （2000年）的计算。 深万科（0002 ）简介： 公司名称：万科企业股份有限公司 A 上市日期：1991-01-29 股 398711877 股，B 股 121755136 股共110504928 股，股权合计数：630971941 股 一、税后净营业利润（NOPAT ）的计算 1 .以表格列出的计算步骤 下表中，最左边一列（以IS 开头）代表损益表中的利润计算步骤， 最右边一列（以NOPAT 开头）代表计算EVA 所用的税后净营业利 润（NOPAT ）的计算步骤。空格代表在计算相应指标（如NOPAT ） 的步骤中不包含该行所对应的 公司简称：深万科 上市地点：上海证券交 易所 行业：房地产业 股本结构：A 股，国有股、境内法人 项。 = 总利润/税前营业利润 -EVA税收调整* - 少数股东权益 = 净利润/税后净营业利润 2.计算公式：（蓝色斜体代表有原始数据，紫色下划线代表此数据需 由原始数据推算出） （1）税后净营业利润二主营业务利润+其他业务利润+当年计提或冲销的坏帐准备一管理费用一销售费用+长期应付款，其他长期负债和住房公积金所隐含的利息 +投资收益一EVA税收调整 注：之所以要加上长期应付款，其他长期负债和住房公积金所隐含的利息是因为sternstewart公司在计算长期负债的利息支出时，所用的长期负债中包含了其实不用付利息的长期应付款，其他长期负债和住房公积金。即，高估了长期负债的利息支出，所以需加回。 （2）主营业务利润=主营业务收入一销售折扣和折让一营业税金及附加一主营业务成本 注：主营业务利润已在sternstewart公司所提供的原始财务数据中直接给出 ⑶EVA税收调整二利润表上的所得税+税率x（财务费用+长期应 付款，其他长期负债和住房公积金所隐含的利息 +营业外支出- 营业外收 目录 1. 均匀分布 ...................................................................................................... 1 2. 正态分布（高斯分布） ........................................................................... 2 3. 指数分布 ...................................................................................................... 2 4. Beta 分布（β分布） .............................................................................. 2 5. Gamma 分布 .............................................................................................. 3 6. 倒Gamma 分布 ......................................................................................... 4 7. 威布尔分布(Weibull 分布、韦伯分布、韦布尔分布) ..................... 5 8. Pareto 分布 ................................................................................................. 6 9. Cauchy 分布（柯西分布、柯西-洛伦兹分布） (7) 10. 2χ分布（卡方分布） (7) 11. t 分布 ......................................................................................................... 8 12. F 分布 ........................................................................................................ 9 13. 二项分布 ................................................................................................ 10 14. 泊松分布（Poisson 分布） .............................................................. 10 15. 对数正态分布 ....................................................................................... 11 1. 均匀分布 均匀分布~(,)X U a b 是无信息的，可作为无信息变量的先验分布。 1 ()f x b a =- Generated by Foxit PDF Creator ? Foxit Software https://www.doczj.com/doc/069920567.html, For evaluation only. 经济增加值(EVA)计算方式(四) 八 下介绍经济增加值的计算方式 下面给出EVA的计算模式。 1、EVA的计算模型 经济附加值= 税后净营业利润—资本成本 = 税后净营业利润—资本总额* 加权平均资本成本 其中： 税后净营业利润= 税后净利润+ 利息费用+ 少数股东损益+ 本年商誉摊销+ 递延税项贷方余额的增加+ 其他准备金余额的增加+ 资本化研究发展费用—资本化研究发展费用在本年的摊销 资本总额= 普通股权益+ 少数股东权益+ 递延税项贷方余额（借方余额则为负值）+ 累计商誉摊销+ 各种准备金（坏帐准备、存货跌价准备等）+ 研究发展费用的资本化金额+ 短期借款+ 长期借款+ 长期借款中短期内到期的部分 加权平均资本成本= 单位股本资本成本+ 单位债务资本成本。 2、报表和账目的调整。 由于根据会计准则编制的财务报表对公司绩效的反映存在部分失真，在计算经济附加值时需要对一些会计报表科目的处理方法进行调整。 Stern Stewart财务顾问公司列出了160多项可能需要调整的会计项目，包括存货成本、重组费用、税收、营销费用、无形资产、货币贬值、坏帐准备金、重组费用以及商誉摊销等。但在考察具体企业时，一般一个企业同时涉及的调整科目不超过15项。但由于EVA是Stern Stewart财务顾问公司注册的商标，其具体的账目调整和运算目前尚没有对外公开。 （1）、单位债务资本成本 单位债务资本成本指的是税后成本，计算公式如下： 税后单位债务资本成本=税前单位债务资本成本*（企业所得税税率） 我国上市公司的负债主要是银行贷款，这与国外上市公司大量发行短期票据和长期债券的做法不同，因此可以以银行贷款利率作为单位债务资本成本。 根据有关研究，我国上市公司的短期债务占总债务的90%以上，由于我国的银行贷款利率尚未放开，不同公司贷款利率基本相同。因此，可用中国人民银行公布的一年期流动资金贷款利率作为税前单位债务资本成本，并根据央行每年调息情况加权平均。不同公司的贷款利率实际上存在一定差别，可根据自身情况进行调整。 （2）、单位股本资本成本 单位股本资本成本是普通股和少教股东权益的机会成本。通常根据资本资产价模型确定，计算公式如下： 普通股单位资本成本=无风险收益+β*市场组合的风险溢价 其中无风险利率可采用5年期银行存款的内部收益率。 国外一般以国债收益作为无风险收益表，我国的流通国债市场规模较小，居民的无风险投资以银行存款为主，因此以5年期银行存款的内部收益率代替。随着国债市场发展，将来也可以国债收益率为基准。 β 系数反映该公司股票相对于整个市场（一般用股票市场指数来代替）的系统风险，β系数越大，说明该公司股票相对于整个市而言风险越高，波动越大。 β值可通过公司股票收益率对同期股票市场指数（上证综指）的收益率回归计算得来。 市场组合的风险溢价反映整个证券市场相对于无风险收益率的溢价，目前有一些学者将我国的市场风险溢价定为4%。 （3）、研究发展费用和市场开拓费用 现行会计制度规定，公司必须在研究发展费用和市场开拓费用发生的当年将期间费用一次性予以核销。这种处理方法实际上否认了两种费用对企业未来成长所起的关键作用，而把它与一般的期间费用等同起来。 这种处理方法的一个重要缺点就是可能会诱使经营者减少对这两项费用的投入，这在效益不好的年份和管理人员即将退休的前几年尤为明显。美国的有关研究表明，当管理人员临近退休之际，研究发展费用的增长幅度确实有所降低。 图 6-2 正态分布概率密度函数的曲线 正态曲线可用方程式表示。当n→∞时，可由二项分布概率函数方程推导出正态分布曲线的方程： f(x)= （6.16 ） 式中： x —所研究的变数； f(x) —某一定值 x 出现的函数值，一般称为概率密度函数（由于间断性分布已转变成连续性分布，因而我们只能计算变量落在某一区间的概率，不能计算变量取某一值，即某一点时的概率，所以用“概率密度”一词以与概率相区分），相当于曲线 x 值的纵轴高度； p —常数，等于 3.14 159 ……； e —常数，等于 2.71828 ……；μ为总体参数，是所研究总体的平均数，不同的正态总体具有不同的μ，但对某一定总体的μ是一个常数；δ也为总体参数，表示所研究总体的标准差，不同的正态总体具有不同的δ，但对某一定总体的δ是一个常数。 上述公式表示随机变数 x 的分布叫作正态分布，记作 N( μ , δ2 ) ，读作“具平均数为μ，方差为δ 2 的正态分布”。正态分布概率密度函数的曲线叫正态曲线，形状见图 6-2 。 （二）正态分布的特性 1 、正态分布曲线是以 x= μ为对称轴，向左右两侧作对称分布。因的数值无论正负，只要其绝对值相等，代入公式（ 6.16 ）所得的 f(x) 是相等的，即在平均数μ的左方或右方，只要距离相等，其 f(x) 就相等，因此其分布是对称的。在正态分布下，算术平均数、中位数、众数三者合一位于μ点上。 2 、正态分布曲线有一个高峰。随机变数 x 的取值范围为（ - ∞，+ ∞ ），在（ - ∞ ，μ）正态曲线随 x 的增大而上升，；当 x= μ时， f(x) 最大；在（μ，+ ∞ ）曲线随 x 的增大而下降。 3 、正态曲线在︱x-μ︱=1 δ处有拐点。曲线向左右两侧伸展，当x →± ∞ 时，f(x) →0 ，但 f(x) 值恒不等于零，曲线是以 x 轴为渐进线，所以曲线全距从 -∞到+ ∞。 4 、正态曲线是由μ和δ两个参数来确定的，其中μ确定曲线在 x 轴上的位置 [ 图 6-3] ，δ确定它的变异程度 [ 图 6-4] 。μ和δ不同时，就会有不同的曲线位置和变异程度。所以，正态分布曲线不只是一条曲线，而是一系列曲线。任何一条特定的正态曲线只有在其μ和δ确定以后才能确定。 5 、正态分布曲线是二项分布的极限曲线，二项分布的总概率等于 1 ，正态分布与 x 轴之间的总概率（所研究总体的全部变量出现的概率总和）或总面积也应该是等于 1 。而变量 x 出现在任两个定值 x1到x2(x1≠x2)之间的概率，等于这两个定值之间的面积占总面积的成数或百分比。正态曲线的任何两个定值间的概率或面积，完全由曲线的μ和δ确定。常用的理论面积或概率如下： 区间μ ± 1 δ面积或概率 =0.6826 μ ± 2 δ =0.9545 μ ± 3 δ=0.9973 μ± 1.960δ=0.9500 μ ±2.576 δ =0.9900 国资委经济增加值考核细则 （详细资料 https://www.doczj.com/doc/069920567.html,/flfg/2010-01/22/content_1517096.htm） 一、经济增加值的定义及计算公式 经济增加值是指企业税后净营业利润减去资本成本后的余额。 计算公式： 经济增加值=税后净营业利润-资本成本=税后净营业利润-调整后资本×平均资本成本率 税后净营业利润=净利润+（利息支出+研究开发费用调整项-非经常性收益调整项×50%）×（1-25%） 调整后资本=平均所有者权益+平均负债合计-平均无息流动负债-平均在建工程 二、会计调整项目说明 （一）利息支出是指企业财务报表中“财务费用”项下的“利息支出”。 （二）研究开发费用调整项是指企业财务报表中“管理费用”项下的“研究与开发费”和当期确认为无形资产的研究开发支出。对于为获取国家战略资源，勘探投入费用较大的企业，经国资委认定后，将其成本费用情况表中的“勘探费用”视同研究开发费用调整项按照一定比例（原则上不超过50%）予以加回。 （三）非经常性收益调整项包括 1.变卖主业优质资产收益：减持具有实质控制权的所属上市公司股权取得的收益（不包括在二级市场增持后又减持取得的收益）；企业集团（不含投资类企业集团）转让所属主业范围内且资产、收入或者利润占集团总体10%以上的非上市公司资产取得的收益。 2.主业优质资产以外的非流动资产转让收益：企业集团（不含投资类企业集团）转让股权（产权）收益，资产（含土地）转让收益。 3.其他非经常性收益：与主业发展无关的资产置换收益、与经常活动无关的补贴收入等。 （四）无息流动负债是指企业财务报表中“应付票据”、“应付账款”、“预收款项”、“应交税费”、“应付利息”、“其他应付款”和“其他流动负债”；对于因承担国家任务等原因造成“专项应付款”、“特种储备基金”余额较大的，可视同无息流动负债扣除。 （五）在建工程是指企业财务报表中的符合主业规定的“在建工程”。 三、资本成本率的确定 （一）中央企业资本成本率原则上定为5.5%。 （二）承担国家政策性任务较重且资产通用性较差的企业，资本成本率定为4.1%。 （三）资产负债率在75%以上的工业企业和80%以上的非工业企业，资本成本率上浮0.5个百分点。 （四）资本成本率确定后，三年保持不变。 四、其他重大调整事项 发生下列情形之一，对企业经济增加值考核产生重大影响的，国资委酌情予以调整。 （一）重大政策变化； （二）严重自然灾害等不可抗力因素； （三）企业重组、上市及会计准则调整等不可比因素 目录 1.均匀分布 (1) 2.正态分布（高斯分布） (2) 3.指数分布 (2) 4.Beta分布（β分布） (2) 5.Gamma分布 (3) 6.倒Gamma分布 (4) 7.威布尔分布(Weibull分布、韦伯分布、韦布尔分布) (5) 8.Pareto分布 (6) 9.Cauchy分布（柯西分布、柯西-洛伦兹分布） (7) χ分布（卡方分布） (7) 10.2 11.t分布 (8) 12.F分布 (9) 13.二项分布 (10) 14.泊松分布（Poisson分布） (10) 15.对数正态分布 (11) 1.均匀分布 均匀分布~(,) X U a b是无信息的，可作为无信息变量的先验分布。 1()f x b a = - ()2 a b E X += 2 ()()12 b a Var X -= 2. 正态分布（高斯分布） 当影响一个变量的因素众多，且影响微弱、都不占据主导地位时，这个变量很可能服从正态分布，记作2~(,)X N μσ。正态分布为方差已知的正态分布 2(,)N μσ的参数μ的共轭先验分布。 22 ()2()x f x μσ-- = ()E X μ= 2()Var X σ= 3. 指数分布 指数分布~()X Exp λ是指要等到一个随机事件发生，需要经历多久时间。其中0λ>为尺度参数。指数分布的无记忆性：{}|{}P X s t X s P X t >+>=>。 (),0 x f x e x λλ-=> 1 ()E X λ = 2 1 ()Var X λ = 4. Beta 分布（β分布） Beta 分布记为~(,)X Be a b ，其中Beta(1,1)等于均匀分布，其概率密度函数可凸也可凹。如果二项分布(,)B n p 中的参数p 的先验分布取(,)Beta a b ，实验数据（事件A 发生y 次，非事件A 发生n-y 次），则p 的后验分布(,)Beta a y b n y ++-，即Beta 分布为二项分布(,)B n p 的参数p 的共轭先验分布。 10 ()x t x t e dt ∞--Γ=? 1 1()()(1)()() a b a b f x x x a b --Γ+= -ΓΓ ()a E X a b = + 2 ()()(1) ab Var X a b a b = +++ 5. Gamma 分布 Gamma 分布即为多个独立且相同分布的指数分布变量的和的分布，解决的 正态分布的数学期望与方差 正态分布： 密度函数为：分布函数为 的分布称为正态分布，记为N(a, σ2). 密度函数为： 或者 称为n元正态分布。其中B是n阶正定对称矩阵，a是任意实值行向量。 称N(0,1)的正态分布为标准正态分布。 （1）验证是概率函数（正值且积分为1） （2）基本性质： （3）二元正态分布： 其中， 二元正态分布的边际分布仍是正态分布： 二元正态分布的条件分布仍是正态分布： 即（其均值是x的线性函数） 其中r可证明是二元正态分布的相关系数。 （4）矩，对标准正态随机变量，有 （5）正态分布的特征函数 多元正态分布 （1）验证其符合概率函数要求（应用B为正定矩阵，L为非奇异阵，然后进行向量线性变换） （2）n元正态分布结论 a) 其特征函数为： b) 的任一子向量,m≤n 也服从正态分布，分布为其中，为保留B 的第，…行及列所得的m阶矩阵。 表明：多元正态分布的边际分布还是正态分布 c) a,B分别是随机向量的数学期望及协方差矩阵，即 表明：n元正态分布由它的前面二阶矩完全确定 d) 相互独立的充要条件是它们两两不相关 e) 若，为的子向量，其中是，的协方差矩阵，则是，相应分量的协方差构成的相互协方差矩阵。则相互独立的充要条件为＝0 f) 服从n元正态分布N(a,b)的充要条件是它的任何一个线性组合服 从一元正态分布 表明：可以通过一元分布来研究多元正态分布 g) 服从n元正态分布N(a,b),C为任意的m×n阶矩阵，则服从m元正态分布 表明：正态变量在线性变换下还是正态变量，这个性质简称正态变量的线性变换不变性 推论：服从n元正态分布N(a,b)，则存在一个正交变化U，使得是一个具有独立正态分布分量的随机向量，他的数学期望为Ua，而他的方差分量是B的特征值。 条件分布 若服从n元正态分布N(a,b)，，则在给定下，的分布还是正态分布，其条件数学期望： (称为关于的回归) 其条件方差为： （与无关） Excel中的正态分布的密度函数 关于在Excel中的正态分布的密度函数NORMDIST（x，μ，σ，逻辑值）中积累逻辑值取“FALSE”时的图形，在《Excel：正态分布的面积图（积累逻辑值为FALSE）》（地址见【附录】）中简单作了尝试。现为了绘制正态累计分布逻辑值要取“TRUE”。 在Excel中的正态分布的密度函数NORMDIST的语法表达式是： NORMDIST(值，平均数，标准差，积累与否)，其中： x ——“值”，是要求分布的随机变量数值； μ——“平均数”，是分布的算数平均数； σ——“标准差”，是分布的标准差； 逻辑值——“积累与否”，是决定函数的逻辑值，其中： ●如果“积累与否”的逻辑值取“TRUE”（真），则NORMDIST 会返回累计分布函数。如果为了绘制正态累计分布，逻辑值就要取 “TRUE”。 ●如果“积累与否”的逻辑值取“FALSE”（伪），则NORMDIST 会返回正态分布函数的高度。 为了制作正态累计分布面积图，先准备下列数据表格（实际使用的表格中，单元格中都是数据，以下为了说明具体公式，在“工具”-“选项”-“视图”中勾选了“公式”，以便各单元格的具体参数都显示出来，以供参考。实际使用时还应该将这个勾选取消）。下列表格中各列NORMDIST函数中的逻辑值都取“TRUE”： 表1 在A列，准备按自己需要设置自变量数据x，本例从0——100，（A2——A102）。 在F列：B2=NORMDIST(A2,50,5, TRUE)，μ=50，σ=5，一直拖到F102。 在G列：G2=NORMDIST(A2,50,10, TRUE)，μ=50，σ=10，一直拖到G102。 在H列：H2=NORMDIST(A2,50,15, TRUE)，μ=50，σ=15，一直拖到H102。 在I列：I2=NORMDIST(A2,70,8, TRUE)，μ=70，σ=8，一直拖到I102。 先选取I列，选取I2：I102，作二维面积图，如图1所示： 图1 再选取H列，选取H2：H102，作二维面积图，如图2所示：经济增加值(eva)计算方式 (四)(Economic value added (EVA) calculation (four))
经济增加值EVA计算方法
经济增加值EVA的计算方法
标准正态分布的密度函数
16种常见概率分布概率密度函数、意义及其应用
经济增加值eva计算方法
16种常见概率分布概率密度函数、意义及其应用
正态分布概率公式(部分)
图 62正态分布概率密度函数的曲线 正态曲线可用方程式表示。 n 当 →∞时，可由二项分布概率函数方程推导出正态 分布曲线的方程：
fx= （61 ） () .6
式中： x—所研究的变数； fx —某一定值 x出现的函数值，一般称为概率 () 密度函数 （由于间断性分布已转变成连续性分布，因而我们只能计算变量落在某 一区间的概率， 不能计算变量取某一值， 即某一点时的概率， 所以用 “概率密度” 一词以与概率相区分），相当于曲线 x值的纵轴高度； p—常数，等于 31 .4 19……； e— 常数，等于 2788……； μ 为总体参数，是所研究总体 5 .12 的平均数， 不同的正态总体具有不同的 μ ， 但对某一定总体的 μ 是一个常数； δ 也为总体参数， 表示所研究总体的标准差， 不同的正态总体具有不同的 δ ， 但对某一定总体的 δ 是一个常数。 上述公式表示随机变数 x的分布叫作正态分布， 记作 N μ ,δ2 )， “具 ( 读作 2 平均数为 μ，方差为 δ 的正态分布”。正态分布概率密度函数的曲线叫正态 曲线，形状见图 62。 （二）正态分布的特性
1、正态分布曲线是以 x μ 为对称轴，向左右两侧作对称分布。因 =
的
数值无论正负， 只要其绝对值相等， 代入公式 61 ） （ .6 所得的 fx 是相等的， () 即在平均数 μ 的左方或右方，只要距离相等，其 fx 就相等，因此其分布是 () 对称的。在正态分布下，算术平均数、中位数、众数三者合一位于 μ 点上。经济增加值(EVA)计算方式 (四) 八
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