advanced_calculus_with_applications_in_statistics_ch6

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洪永淼老师统计学第三章

Chapter3Random V ariables and UnivariateDistributionsKey words:continuous random variable,cumulative distribution function,discrete random variable,probability density function,probability mass function,random variable,moments, moment generating function,mean,variance,skewness,kurtosis.Abstract:In this and next chapters,we will use advanced calculus to formalize the probability theory introduced in Chapter2.The use of mathematics enables us to investigate probability more deeply.A number of quantitative-oriented probability concepts will be introduced.In this chapter,we…rst introduce the concept of a random variable and characterize the distributions of a random variable and functions of a random variable by the cumulative distribution function,the probability mass function or probability density function,and the moment generating function and the characteristic function.We also introduce a class of moments as well as their relationships with a distribution.This chapter focuses on univariate distribution.3.1:Random Variables and Distribution FunctionsRead Sections1.4–1.6Review:Recall that B;Borel…eld(or -algebra),is a collection satisfying the following conditions:(i) 2B;(ii)If A2B then A c2B;(iii)If A j2B;j=1;2;:::;then[1j=1A j2B:When B is a -algebra,we call the pair(S;B)a measurable space.Also,a probability function is a mapping from B to R satisfying the following conditions(i).P(A) 0for all A2B;(ii).P(S)=1;(iii).If A j2B;j=1;2; ;are pairwise mutually exclusive,then P([1j=1A j)=P1j=1P(A j): De…nition[Probability Space]:Suppose P is a probability measure de…ned on the measurable space(S;B).Then(S;B;P)is a probability space.Remark:The space S and the associated -algebra di¤er according to the natures of random experiments.Example:A Coin is thrown.S=f H;T g:Example:Bush running for the next term of Presidency.S=f Win,Fail}.Example:Three coins are thrown.S=f HHH;HHT;HT H;HT T;T HH;T HT;T T H;T T T g.: Event A={two heads appear}=f HHT;HT H;T HH g:Then P(A)=38Example:Throwing a die,S=f1;2;3;4;5;6g.Remark:It is inconvenient to work with di¤erent sample spaces.In particular,a sample spaceS may be tedious to describe if the elements of S are not numbers.In many experiments,it is easier to deal with a summary variable than with the original probability structure.To developa uni…ed probability theory,we need to unify di¤erent sample spaces.For this purpose,we needto formulate a rule,or a set of rules,by which elements of S may be represented by numbers. This can be achieved by assigning a real number to each possible outcome in S.In other words, we construct a mapping from the original sample space S to a new sample space ,a set of real numbers.This transformations is called a random variable.A random variable is a function de…ned on a sample space.Its purpose is to facilitate the solution of a problem by transferring considerations to a new probability space with a simpler or more convenient structure.On the other hand,in many applications,we are interested only in a particular aspect of the outcomes of experiments,rather than all outcomes.For example,when we roll a number of dice, we are usually interested in the total,and not in the outcome of each dice.In such applications,a suitably de…ned random variables will serve for our purpose.De…nition[Random Variable]A random variable(r.v.)X( )is a B-measurable mapping(or point function)from the sample space S to the real line R such that to each outcome s2S there exists a corresponding unique real number,X(s):The collection of all possible values that the random variable X can take,also called the range of X( );constitutes a new sample space, denoted as .Example:S=f H;T g.De…ne X(H)=1and X(T)=0:Then =f1;0g:Example:S=f Win,Fail}:De…ne X(W in)=1and X(F ail)=0:Then =f1;0g:Remark:It is not necessary to have the same number of basic outcomes for S and :In some cases,it is more convenient to work with a suitably de…ned new sample space .Example:S=f T T T;T T H;T HT;HT T;HHT;HT H;T HH;HHH g.De…ne X(T;T;T)= 0;X(T;T;H)=1;X(T;H;T)=1;X(H;T;T)=1;X(H;H;T)=2;X(H;T;H)=2;X(T;H;H)= 2;X(H;H;H)=3:Then =f0;1;2;3g:Intuition:X(s)is the number of heads.Therefore,P(X=3)=P(A);where A=f s in S:X(s)=3g;denotes the probability that exactly three heads occur in the experiment.Example:S=f1;2;3;4;5;6g:De…ne X(s)=s:Then =S:Identity transformation.Need not change S because S consists of numbers already.Remark:The transformation need not be1 1.Question:Suppose the number of basic outcomes in S is countable.(a)Is it possible that the number of basic outcomes in is larger than that of S?(b)Smaller than that of S?Example:S=f s: 1<s<1g:X(s)=1if s>0and X(s)=0if s 0:Remark:This is useful for directional forecasts or investigation of asymmetric business cycles (e.g.,Neftci1984,“Are Economic Time Series Asymmetric Over the Business Cycles?”,JPE92, 307-328).Remark:The above de…nition of a random variable is limited to real-valued functions.This does not impose any restriction.We can de…ne complex-valued random variables by looking upon the real and imaginary parts separately as two real-valued random variables.Remark:In this book,we use a capital letter X to denote a random variable,and use a lowercase letter x to denote its realization.In de…ning a random variable X,we have also de…ned a new sample space (the range of the random variable X).The probability function de…ned on the original sample space S can be used for the random variable X.Suppose we have a sample space with…nite basic outcomesS=f s1;:::;s n gwith a probability function P( )and we de…ne a random variable X with the range=f x1;:::;x m g:We can de…ne the probability function P X( )on in the following way:P X(x i) P(X=x i)=P(C);where C is an event in S such thatC=f s2S:X(s)=x i g:Note that the left hand side,P X( );is an induced probability function on ;de…ned in terms of the original probability function P( ):This can be applied to the case where is countable. For an uncountable ;we can still de…ne the induced probability function,P X( );in a similar manner.More generally,for any set A2B ;where B is a Borel…eld generated from ;P X(A) P(X2A)=P(C)=P[s2S:X(s)2A];where C=f s2S:X(s)2A g is the set of basic outcomes in the original sample space S for which the random variable X has a value that is in A:Thus,a random variable X is a function that carries the probability from a sample space S to a space of real numbers.In this sense, with A2 ;the probability P X(A)is often called an induced probability function.It can be shown that when S is a countable sample space,the induced probability function P X( )satis…es the three axioms of the probability function.Remark:When S is continuous(e.g.S=R);we cannot use the above expressions immediately unless we can be sure that the set C=f s2S:X(s)2A g belongs to the -algebra B.Whether or not C2B depends on,of course,the form of mapping X( ).Question:What functional form X( )will ensure C2B?Or what condition on X( )will ensure that the set C is in B?The following condition ensures that C always belongs to B:De…nition[Measurable Function]:A function X:S!R is B-measurable(or measurable with respect to the sigma…eld B generated from S)if for every real number a;the set[s2S: X(s) a]2B:Dice S=f1;2;3;4;5;6g;=S;X(s)=s:Remarks:(i)When a function X is B-measurable,we can express the probability of an event,say "X a";in terms of the probability of event C in B;where C=f s2S:X(s) a g:In other words,the measurable function ensures that P(X2A)is always well-de…ned for all subsets A on B .If X( )is not a measurable function,then there exist subsets on the Boreal…eld in R for which probabilities are not de…ned.However,constructing such sets are very complicated.(ii)In what follows,we assume that the random variable X is always measurable with respect to some -algebra of S.In fact,in the advanced probability theory,the term“random variable”is restricted to be a B-measurable function from S!R:Theorem:Let B be a -algebra associated with sample space S.Let f and g be B-measurable real valued functions,and c be a real number.Then the functions c f;f+g;f g and j f j are also B-measurable.Proof:See White[1984,Proposition3.23],who says“see Bartle[1966,Lemma2.6]”. Remark:If one starts with function X(s)and Y(s)that are measurable mappings from S to ;then new functions Z(s)can be constructed by ordinary algebraic operations such as Z(s)=aX(s);Z(s)= X(s)+Y(s);Z(s)=X(s)Y(s);and Z(s)=X(s)=Y(s)are measurable.If one has a sequence X1(s);X2(s);:::of measurable functions and constructs Z(s)through limiting operations such as Z(s)=lim i!1Z i(s)or Z(s)=sup1 i<1f Z i(s)g;then Z(s)is measurable.A proof of these claims is not particularly di¢cult,but is beyond this course.All that is important for our purposes is to know that standard functions are measurable and that any standard sequence of countable operations on such functions will not destroy measurability.It can be shown that the induced probability P X( )satis…es the de…nition of a probability function.First,conditions(i)and(ii)are easily veri…ed by observing,for C=f s2S:X(s)2 A g;thatP X(A)=P(C) 0;and that S=f s2S and X(s)= g requiresP X( )=P(S)=1:In discussing condition(iii),let us restrict our attention to two mutually exclusive events A1and A2in :Here,P X(A1[A2)=P(C);where C=f s2S:X(s)2A1[A2g:However,we haveC=f s2S:X(s)2A1g[f s2S:X(s)2A2g;or for brevity,C=C1[C2:But C1and C2are disjoint sets in S.This must be so,for if some s were common,say s0;then X(s0)2A1and X(s0)2A2:That is,the same number X(s0) belongs to both A1and A2.This is a contradiction because A1and A2are disjoint sets in . Accordingly,P(C)=P(C1)+P(C2):However,by de…nition,P(C1)=P X(A1)and P(C2)=P X(A2);and thusP X(A1[A2)=P X(A1)+P X(A2):This is the condition(iii)for two disjoint sets when de…ning a probability function. Remark:The reason that we need the random variable X to be a Borel measurable function is to assure that we can…nd the induced probabilities on the sigma…eld B X generated from the subsets of :We need this requirement throughout this course for every function that is a random variable.Remark:P X( )is the probability function de…ned on the Borel…eld B X generated from :In many cases below,we will drop the subscript X and simply write P X(A)=P(A)for A B X:Example(Three Coins thrown,continued):S=f HHH;:::;T T T g; =f0;1;2;3g: P[0 X 1]=P f s2S:0 X(s) 1g:A=f s2S:0 X(s) 1g=f T T T;T T H;T HT;HT T g:P[0 X 1]=P(A)=1=8+1=8+1=8+1=8:Question:How to characterize a random variable X?Answer:Of course,we can use the probability function P X(A):Below,we introduce an alter-native function to characterize the distribution of X:Cumulative Distribution Function(CDF)of a random variable X is de…ned as:F X(x)=P(X x)for all x2R:Remark:The subscript X of the F function indicates that it is the CDF of the random variable X:Properties of F X(x):(i)lim x! 1F X(x)=0;lim x!+1F X(x)=1:(ii)F X(x)is non-decreasing,i.e.for any x1<x2;F X(x1) F X(x2).(iii)F X(x)is right-continuous,i.e.for all x and >0;lim!0+[F X(x+ ) F X(x)]=0:Implications and interpretations:0 F X(x) 1:Theorem:Let a<b:ThenP(a<X b)=F X(b) F X(a):Theorem:P(X>b)=1 F X(b):Example:If F(x)is a cumulative distribution function(cdf),is G(x)=1 F( x)a cdf too? Solution:(i)G( 1)=1 F[ ( 1)]=1 F(+1)=1 1=0;G(+1)=1 F( 1)=1 0=1:(ii)x1<x2:G(x1)=1 F( x1);G(x2)=1 F( x2):G(x2) G(x1)=[1 F( x2)] [1 F( x1)]=F( x1) F( x2)0:(iii)No.G(x)is left-continuous but not necessarily right-continuous.Example:Is P(X b)=1 F X(b)?Solution:P(X b)=P(X>b)+P(X=b)=1 F X(b)+P(X=b):Example:Suppose F1(x)and F2(x)are two cumulative distribution functions,isF(x)=pF1(x)+(1 p)F2(x)a cdf,where p is a constant.Remark:In application,p may depend on economic variables.An example of p=p(Z)isp(Z)=11+exp( 0Z);where Z is the state variable of economy,taking0with prob p and taking1with prob 1 p:This is a mixture distribution.A mixture of two distributions can provide a great deal of ‡exibility.Answer:Yes if0 p 1:Remark:When proper conditions on p are imposed,this is called a mixture of distributions.A well-known example in econometrics is the so-called Markov regime-switching model in which p depends on a state variable characterizing the business cycles(cf.Hamilton1994,Time Series Analysis).Remark:The de…nition of the cdf makes it clear that the probability function P X( )determines the cdf F X( ):It is true,although not so obvious,that a probability function P X( )can be found from a cdf F X( ):That is,P X( )and F X( )give the same information about the distribution of probability,and which function is used is a matter of convenience.Question:Why does the CDF F X(x)can characterize the probability distribution of random variable X?De…nition:Two random variables X and Y are identically distributed if for every set in B1;where B1is the smallest -…eld containing all the intervals of real numbers of the form (a;b);[a;b);(a;b];and[a;b];one hasP(X2A)=P(Y2A):Remark:Identical distribution does not imply X=Y:Theorem:Two random variables X and Y are identically distributed if and only ifF X(u)=F Y(u)for all 1<u<1:3.2Discrete Random Variable(D.R.V.)Read Sections3.1and3.2De…nition[Discrete v.s.Continuous Random Variables(R.V.)]:If a random variable X can only take a countable number of values,then X is called a discrete random variable(d.r.v.). Otherwise,if it can take any value in an interval,it is called a continuous random variable(c.r.v.). Examples:=f1;2;3;4;5;6g:=f0;1;2;3;::::g;e.g.,Poisson distribution.=R:=[0;1]:Prob(Mass)Function(pmf)of a discrete r.v.X is de…ned asf X(x)=P(X=x)for all x2R:Properties of pf/pmf:(i)0 f X(x) 1for all x2R:(ii)P x f X(x)=1:De…nition[Support of X]:The collection of the points at which a d.r.v.X has a positive probability is called the support of X;denoted asSupport(X)=f x2R:f X(x)>0g:Remark:Although f X(x)is de…ned on x2R;it su¢ces to know the support of a d.r.v.X and the probabilities of all points in the support.Geometric representation:Probability histogram.For example,it can show whether there are more than one mode,or“high points”.A mode is a bar in a histogram that is surrounded by bars of lower frequency.A histogram exhibiting two modes is said to be bimodal,and one having more than two modes is said to be multimodal.Question:Given the probability function,What can we obtain?Cumulative Distribution Function of a d.r.v.X isF X(x)=P(X x)=X y x f X(y)for any x2R;where the summation is over all values y that are less than or equal to x.Remark:The argument of F X( )runs continuously from 1to+1;i.e.x2( 1;1):It is di¤erent from the support of X:Example1:Uniform Distribution U(N);X12f X1=N1=NFind the distribution function.Solution:Case1:x<1.Then f X x g is ;andF X(x)=X x i x f X(x i)=0:How many x0i s that are less than or equal to x?“X x"= :Case 2:1 x <2.Then “X x ”=f 1g ;andF X (x )=X x i xf X (x i )=f X (1)=1=N:Case 3:2 x <3:Then “X x ”=f 1;2g ;and F X (x )=2=N:Case j :j 1 x <j;j N:Then “X x ”=f 1;2;3; ;j 1g ;andF X (x )=(j 1)=N:Case N +1:x N:Then"X x "=f 1;2;3; ;N g ;andF X (x )=1:In summary,F X (x )=8<:0x <1j=N j x <j +1;1 j <N;j 2Z 1x NRemark:F X (x )is a function on R ;so it is incomplete if you just compute F X (x )on x =1;2;3;4; ;N:Conclusion:Given f X (x );we can obtain F X (x ):Question :Suppose X is a d.r.v.that takes values of x 1<x 2<x 3< ;and its distribution function F X (x )is given,is it possible to recover f X (x )from F X (x )?ANS:For d.r.v.,F X (x )is a step function.The points f x i g where f x (x i )>0will be the jump points for F X (x ).Without loss of generality,we can arrange these points in an increasing sequential order.Then the probabilities at these points can be obtained fromf X (x i )=P (X =x i )=P (x i 1<X x i )=F X (x i ) F X (x i 1)for i =2;3;:::For i =1;we have f X (x 1)=F X (x 1)because f X (x 1)=P (X =x 1)=P (X x 1)=F X (x 1)given that x 1is the smallest value that X can take.f X =x 1g =f X x 1g Remark:f X (x )and F X (x )are equivalent ways to describe the discrete random variable X .Given f X (x )or F X (x );we can know “everything”of the probability law of X .Examples of D.R.V.Example[Bernoulli R.V.]:A d.r.v.X is called a Bernoulli(p)random variable if its pmff X(x)= p if x=11 p if x=0;where0<p<1:Remark:This distribution arises when one tosses a coin whose head shows up with probability p:Example[Binomial R.V.]:A d.r.v.X is called a Binomial(n;p)if its pmff X(x)=(n x)p x(1 p)n x;x=0;1;:::;n;where n 0and0<p<1:Remarks:When can this distribution arise?A person throws a coin n times independently. Each time the head has probability p and the tail q=1 p:How many heads can he get from these n trials?Interpretation:X=P n i=1X i;where X i is a sequence of independently identically distribu-tion(i.i.d.).Bernoulli random variable(p):Question:How to verifynX x=0(n x)p x(1 p)n x=1for all p2(0;1)?ANS:By the binomial theorem which states that for any real numbers x and y and integern 0;(x+y)n=nX i=0(n i)x i y n i:See p.90of the textbook.In our application,we put x=p and y=1 p:Question:Why is the binomial distribution useful?Example[Discrete Uniform Distribution]A d.r.v.X is a discrete uniform(1;N)distribution iff X(x)= 1N x=1;:::;N;0otherwise,where N is a speci…ed positive integer.Example[Poisson Distribution]A d.r.v.X follows a Poisson( )distribution if its pmff X(x)= e x x!;x=0;1;:::;0otherwise,where >0:The parameter is called an intensity parameter.Question:What is the interpretation of ?Answer:It is proportional to an instantaneous probability.Question:How to verify1X x=0e x x!=e 1X x=0 x x!=1?Hint:What is the Taylor series expansion of e y?For any y near zero,1X i=0y i i!e y=Put i=x;y= :Remark:Poisson distribution is very popular in…nance.It has been used to capture jumps in …nancial markets.3.3Continuous Random VariablesRead Section3.3De…nition[Continuous Random Variable]A r.v.X is called continuous if its distribution function F X(x)is continuous for all x.Alternatively,a r.v.X is discrete if F X(x)is a step function of x:Picture:F X(x)is a continuous function.A c.r.v.X provides a proper description of income, temperature,stock return,and so on.Question:Can we de…ne a pmf f X(x)for a continuous random variable X?Obviously,for any constant">0;0 P(X=x)P(x "=2<X x+"=2)=F X(x+"=2) F X(x "=2)!0as"!0:Thus,P(X=x)=0for all x:That is,if X is a continuous random variable,the probability that X takes a single point is zero.This has some important implications.For example,for a c.r.v.,we haveP(a<X b)=P(a X<b)=P(a X b):Question:Under what conditions,can we write F X(x)=R x 1f X(y)dy for some function f X(x)? De…nition[Absolute Continuity(AC)]:A function F:R!R is called absolutely contin-uous with respect to Lebesgue measure if F(x)is continuous on R and is di¤erentiable almost everywhere(i.e.for almost all x):Remark:What is meant by“almost everywhere”?Intuitively,in any…nite interval of R;there is a…nite number of points or an in…nite but countable number of points where F X(x)is not di¤erentiable.De…nition[Probability Density Function(pdf)]:Suppose the distribution function F X(x) of a c.r.v.X is absolutely continuous.Then there exists a function f X(x)such thatF X(x)=Z x 1f X(u)du for all x2( 1;1).The function f X(x):R!R is called a probability density function of X.Question:Is f X(x)a unique function for a given F X(x)?Remarks:(i)When the AC condition does not hold,X may not have the above relationship.Throughout this course,we will assume that F X(x)is AC.(ii)For those x’s where F0X(x)exists,we have from the above thatf X(x)=dF X(x) dx:Given that F X(x)is AC,the probability density function f X(x)exists almost everywhere. Question:How to de…ne f X(x)at the points where F0X(x)does not exist?Answer:Arbitrary but try to make f X(x)as smooth as possible.For a c.r.v.X with well-de…ned pdf f X(x);f X(x)and F X(x)are equivalent to each other. Given one,we can always recover the other.(iii)[Interpretation of pdf]:For any small constant">0;we have,by the mean value theoremP[x "=2<X x+"=2]=F X(x+"=2) F X(x "=2)=Z x+"=2x "=2f X(y)dy=f X( x)"for some x2[x "=2;x+"=2](by the mean-value theorem)=probability that X is between(x "=2;x+"=2]: Although f X(x)is not a probability measure,it is proportional to the probability that X takes values in a small interval centered at x:(iv)The fact that P(X=0)=0for a c.r.v.allows us to change the value of the pdf of a continuous random variable X at a single point without altering the distribution of X:For instance,the pdff X(x)= e x for0<x<1;0elsewhere,can be written asf X(x)= e x for0 x<1;0elsewhere,without changing P X(A)for any A in R:We observe these two functions di¤er only at x=0 and P(X=0)=0:More generally,if two pdf’s of continuous random variables di¤er only on a set having probability zero,the two corresponding probability set functions are exactly the same.Unlike the continuous random variables,the pmf of a discrete random variable may not be changed at any point,since a change in such a pmf alters the distribution of probability.(v)Because f X(x)is a slope that determines the change of a probability F X(x),f X(x)can be greater than1.(vi)Remark:Because of the property that P(X=x)=0for all x2R;the values of any pdf f X(x)can be changed at a…nite number of points or even at an in…nite sequence of points, without changing the value of the cdf F X(x):In other words,the values of the pdf of X can be changed arbitrarily at an in…nite sequence of points without a¤ecting any probabilities involving X;that is,without a¤ecting the probability of X:To the extent just described,the pdf is not unique.In many problems,however,there will be one version of the pdf that is more natural than any other pdf.For this reason,the pdf will, wherever possible,be continuous on the real line.Theorem[Properties of pdf]:A function f X(x)is a pdf of a c.r.v.X i¤(a)f X(x) 0for all x and(b)R1 1f X(x)dx=1:Proof:[Necessary]Suppose f X is a pdf,i.e.F X(x)=R x 1f X(y)dy:By the mean value theorem,F X(x+ ) F X(x)=f X( x) ;where x lies between x and x+ :It follows that f X( x) 0since F X( )is nondecreasing.Also,(b)follows because F X(1)=1:[Su¢ciency]The converse can be proved similarly by constructing F X(x)=R x 1f(y)dy and showing that F X(x)is a cdf of some random variable X.First,F X(x)is nondecreasing by the mean valued theorem and(a),F X(1)=1by(b),and F X( 1)=0by(a)and(b).Also, F X(x)=R x 1f(y)dy implies that F X(x)is continuous,becauseF X(x+ ) F X(x)=Z x+ x f(y)dy!0as !0;and so is right-continuous.Therefore,F X(x)is a cdf of some random variable X.Geometric Representation of pdf f X(x):Indicative of mode,skewness or symmetry,heavy tails,and etc.Questions:(i)Is it possible to construct a pdf f(x)from any nonnegative function g(x)with a…nite integral(i.e.,0<R1 1g(x)dx<1)?ANS:Yes,f(x)=g(x)=R1 1g(y)dy:An application in…nance:[stochastic discount factor and risk neutral probability density function]In a dynamic setting,…nancial derivatives prices are determined by the Euler equationE[M t;t+ Y(S t+ )j I t]=P t;where M t;t+ 0is the so-called stochastic discount factor of the representative economic agent, Y(S t+ )is the pay o¤scheme at the maturity date t+ ;S t+ is the price of the underlying asset (e.g.,Standard&Poor500closing price index)at time t+ ;P t is the price of the derivative at time t,and I t is the information available at time t:We can write the Euler equation equivalently as follows:ZM t;t+ Y(s)f t+ (s)ds=P t;where f t+ (s)is the pdf of S t+ conditional on I t:Now de…ne a so-called risk neutral pdff (s t+ j I t)=M t;t+ f t+ (s)E[M t;t+ j I t]=M t;t+ f t+ (s) R M t;t+ f t+ (s)ds:Then the Euler equation can be written ase r E [Y(S t+ )j I t]=P t;ore r Z Y(s)f t+ (s)ds=P t;where r is the compound interest rate,and E ( j I t)is the so-called expectation under the risk neutral pdf f t+ (s):(ii)Is it possible to construct a symmetric pdf f(x)from any nonnegative function g(x)with a…nite integral?Remark:symmetric if f(x)=f( x)for all x:ANS:f(x)=g(x)+g( x)2R1 1g(y)dy:De…nition[Support]:The support of a c.r.v.X is de…ned asSupport(X)=f x2R:f X(x)>0g;where f X(x)is the pdf of X:Remark:It su¢ces to focus on the support of X:Examples of c.r.v.Example [Uniform Distribution]A c.r.v.X follows a uniform distribution on [a;b ]if its pdff X (x )= 1b a a x b 0otherwise,where a <b:Example [Gamma Distribution]a c.r.v.X follows a Gamma ( ; )distribution iff X (x )= 1 ( ) x 1e x= 0<x <10otherwise,where ; >0:Question:What is the gamma function ( )?( )=Z 10t 1e t dt:Properties of ( ):(i) ( +1)= ( );(ii) (n )=(n 1)!if n is a positive integer.(iii) (12)=p :Question:How to verifyZ 10f X (x )dx =Z 101 ( )x 1e x= dx =1?Hint:By change of variable and de…nition of the gamma function.Z 1 1f X (x )dx =Z 101 ( ) x 1e x= dx =1 ( )Z 10(x= ) 1e (x= )d (x= )(set y =x= )=1 ( )Z 10y 1e y dy =1 ( ) ( )=1:Remark:The Gamma distribution has been popularity used to model the waiting time of economic events.Example[Exponential Distribution]X follows Exponential( )iff X(x)= 1 e x= 0<x<10otherwise,where >0:Remarks:(i)Exponential( )=Gamma(1; ):(ii)Exponential Distribution is popular in modelling duration between…nancial events or economic events.An example in labor economics:Let T be the unemployment duration of a worker which has a pdf f T(t).Then the so-called hazard rate is de…ned as(t)=limt!0+P(T t+ t j T t)t=f T(t)P(T t);which is the instantaneous probability that the worker will…nd a job after an umemployment duration of t:A simplest example is to assume thatf(t)=1e t= for t>0:Then the hazard rate will become(t)=1 ;which is a constant of time.(iii)An empirical stylized fact of high-frequency…nancial return f X t g is that j X t j approxi-mately follows an exponential distribution(see Ding,Granger and Engle1993,Journal of Em-pirical Finance.)Here,X t is the standardized…nancial return.Example[Normal Distribution]A normally distributed random variable,X N( ; 2);has thepdff X(x)=1p2 e (x )22 2; 1<x<1;where 1< <1and0< <1:Remark:A normal distribution is also called Gaussian distribution.Why?Answer:The normal distribution was discovered in1733by Abraham De Moivre(1667-1754)in his investigation of apprxomating coin tossing probabilities.he named the PDF of his discovery the exponetially bell-shaped curve.In1809,Carlk Friedrich Gauss(1777-1855)…rmly established the importance of the normal distribution by using it to predict the location of astronmical bodies.As a result,the normal distribution then became commonly known as the Gaussian distribution.Remarks:(i)X is called standard normal,denoted X N(0;1);if =0and =1:(ii)Normal distribution is the most important distribution in probability theory.The Central Limit Theorem(CLT)says that under suitable conditions,a sample average will converge to a normal distribution as the sample size increases.1 p nnX i=1X i!Normal distribution as n!1.This follows no matter whether X i is discrete or continuous,is compact supported or uncompact supported.(iii)Question:How to verifyZ1 11p2 exp (x )22 2 dx=1?Put y=(x )= :Then it becomesZ1 11p2 exp y22 dy=1:Question:How to verify this?Z1 1e x22dx 2=Z1 1e x22dx Z1 1e y22dy=Z1 1Z1 1e x2+y22dxdy(set x=r cos( );y=r sin( ))=Z10Z2 0e r22rdrd=2 Z10e r22rdr=2 :It follows that1p2 Z1 1e x22dx=1:Example[Beta distribution]X follows a Beta( ; )distribution iff X(x)= 1B( ; )x 1(1 x) 10<x<1;0otherwise,。

数学选必一第一章知识点

数学选必一第一章知识点Mathematics is a foundational subject that plays a crucial role in various aspects of our daily lives. From basic arithmetic to advanced calculus, mathematics shapes our understanding of the world around us and provides a framework for problem-solving and critical thinking. In the first chapter of the math elective, students delve into a range of fundamental concepts that lay the groundwork for more complex topics to come.数学是一个基础性的学科，对我们日常生活的各个方面起着至关重要的作用。

从基础的算术到高级的微积分，数学塑造了我们对周围世界的认识，为问题解决和批判性思维提供了框架。

在数学选必一的第一章中，学生们深入了解了一系列基本概念，为以后更复杂的主题奠定了基础。

One of the key topics covered in the first chapter of the math elective is number theory. This branch of mathematics focuses on the properties and relationships of numbers, exploring concepts such as divisibility, prime numbers, and modular arithmetic. Through studying number theory, students develop a deeper understandingof the fundamental building blocks of mathematics and how numbers interact with each other.数学选必一第一章涵盖的一个关键主题是数论。

经历挫折后成功的英语作文

经历挫折后成功的英语作文全文共3篇示例，供读者参考篇1Hitting Rock Bottom Before Soaring HighAs I raise my eyes to look back on my journey through high school, I can't help but be struck by the immense growth I've experienced, both academically and personally. If you had told the young, naive freshman version of myself that I would one day be where I am now, I would have scoffed in utter disbelief. My path has been riddled with obstacles that seemed insurmountable at times, causing me to stumble and fall harder than I ever thought possible. However, it was precisely those setbacks that ultimately fueled my determination and propelled me towards the greatest achievements of my life thus far.My first major hurdle came in the form of my sophomore year chemistry class. I had always excelled in science, so I naively assumed chemistry would be a walk in the park. Boy, was I wrong. From the get-go, I struggled to grasp even the most fundamental concepts, and my test scores reflected my utter bewilderment. I Can vividly recall the sinking feeling in the pit ofmy stomach as I held my first chemistry exam only to find a dismal 52% scrawled atop the page in harsh red ink.In that moment, I had a crucial decision to make: I could either resign myself to failure or pour every ounce of effort into turning my fortunes around. Thankfully, I chose the latter. I started attending extra help sessions religiously, poring over practice problems until my eyes grew weary. Slowly but surely, the fog began to lift, and chemistry started making sense to me in a way it never had before.My perseverance paid off, and by the end of the year, I had clawed my way back to earn a respectable B in the class. More importantly, though, I had learned a lesson that would prove invaluable moving forward: With enough grit and determination, even the most monumental obstacles can be overcome.If only the challenges ended there. As I transitioned into my junior year, I found myself in the throes of a different kind of crisis – one that shook me to my core. My parents, whose marriage had been fraying for years, finally made the wrenching decision to separate. Needless to say, this sent my world into a tailspin. Focusing on schoolwork suddenly felt like an insurmountable task as I grappled with the emotional turmoil of my family's breakdown.My grades began to plummet as achievement took a backseat to anguish. I distinctly remember huddling in a restroom stall during my free period, racked with sobs as the full weight of my circumstances crushed down on me. In that moment of despair, the notion of graduating – let alone getting into college – seemed laughably unrealistic.And yet, as the weeks churned on, a quiet resilience began to take root within me. I realized that despite my circumstances, I still held the power to shape my own destiny. While I couldn't control the fracturing of my family, I could control how I responded to that adversity. I could either crumple in the face of hardship or use it as fuel to forge myself into a tougher, more resilient person.With a newfound determination, I slowly but surely brought my academic performance back to par. I sought counseling to help process my emotions in a healthy way, and invested more time than ever before into my studies. My grades gradually improved, and by the end of my junior year, I had regained the academic footing that had once seemed so immutable.As senior year took hold, I found myself galvanized by a palpable sense of vigor. I had traversed the deepest of valleys,and emerged bathed in hard-won perseverance and self-belief. No challenge seemed too formidable to take on any longer.It was with this steadfast mindset that I tackled the herculean gauntlet of college applications. Each essay, each form, was approached with a zeal and diligence that would have been unrecognizable to my former self. When Decision Day finally arrived, I held my breath as I opened email after email – until finally, there it was in digital ink: My acceptance to my dream school.In that moment, the weight of my tribulations came crashing down in cathartic waves of euphoria. I had been forged in the fires of adversity, emerging from the flames like a phoenix –emboldened, tempered, and indomitable.As I prepare to bid farewell to the hallowed halls of my high school, I carry with me not only a formidable academic transcript, but the lasting knowledge that no obstacle is truly insurmountable if met with unstoppable determination. My failures enabled my successes, sculpting me into the resolute individual I am today.While the road ahead will undoubtedly present new challenges to confront, I face the future with a groundedself-assurance. I have scaled mountainous peaks of adversity,and lived to tell the tale. Whatever comes my way, I know in my heart that I possess the grit and fortitude to persevere.For the fallen shall be given wings to soar higher than ever before imagined. That is the hard-earned truth etched into the fabric of my soul after this winding journey we call adolescence. While the trials were harsh, the lessons were invaluable – and for that, I will be eternally grateful.篇2My Journey of Failure and TriumphAs I sit here reminiscing about my academic journey, a multitude of emotions washes over me. The road has been paved with challenges, setbacks, and moments that tested the very fabric of my resilience. However, it is through these trials that I have emerged stronger, wiser, and more determined than ever before.The first significant hurdle I encountered was during my freshman year of high school. I had always been a diligent student, but the transition to a more rigorous curriculum proved overwhelming. My grades began to slip, and I found myself struggling to keep up with the pace of the coursework.Frustration and self-doubt crept in, threatening to derail my academic aspirations.It was during this tumultuous time that I learned a valuable lesson: failure is not a permanent state but a temporary setback. With the unwavering support of my parents and the guidance of my teachers, I developed a study plan that catered to my learning style. Late nights were spent poring over textbooks, seeking clarification from tutors, and honing mytime-management skills.Slowly but surely, my hard work paid off. My grades began to improve, and a newfound confidence took root within me. I had faced adversity head-on and emerged victorious, armed with the knowledge that perseverance and dedication are the keys to overcoming any obstacle.As I progressed through high school, I encountered numerous challenges, each one shaping me into the person I am today. From navigating the complexities of advanced courses to juggling extracurricular activities and part-time jobs, I learned the art of prioritization and the importance of maintaining a healthy balance.One particularly daunting challenge was preparing for the college admission process. The pressure to excel academically,craft the perfect personal statement, and secure glowing recommendations was immense. Despite my best efforts, I faced rejection from several prestigious institutions, leaving me disheartened and questioning my abilities.It was during this low point that I realized the true value of resilience. Instead of wallowing in self-pity, I decided to channel my energy into exploring alternative paths. I researched schools that aligned with my passions and aspirations, ultimately finding a university that recognized my potential and offered me a place in their esteemed program.The transition to college life was exhilarating yet daunting. Surrounded by a diverse community of scholars, I found myself constantly challenged to think critically, question assumptions, and push the boundaries of my knowledge. It was during this time that I truly embraced the art of learning, recognizing that failure is an integral part of the journey towards success.One particular course stands out as a testament to my perseverance. Advanced Calculus initially seemed like an insurmountable obstacle, with its complex concepts and intricate problem sets. I struggled to grasp the material, and my confidence wavered with each disappointing grade. However, instead of giving up, I sought guidance from my professor,attended extra tutoring sessions, and formed study groups with like-minded peers.Slowly but surely, the fog of confusion lifted, and the concepts began to click. Each small victory fueled my determination, and I found myself approaching the subject with a newfound passion and curiosity. By the end of the semester, I had not only mastered the material but also gained a profound appreciation for the beauty and elegance of mathematics.As I reflect on my academic journey, I am filled with a sense of gratitude for the lessons learned through failure. Each setback served as a catalyst for growth, pushing me to develop resilience, perseverance, and an unwavering determination to succeed.Looking ahead, I am filled with excitement and optimism. The challenges I have faced have forged within me a steely resolve to tackle whatever obstacles lie ahead. I am confident in my ability to adapt, learn, and grow, for I now understand that failure is not the antithesis of success but rather a stepping stone towards greatness.To my fellow students embarking on their own academic journeys, I offer this advice: Embrace failure as an opportunity to learn and grow. Seek guidance from those who have walked the path before you, and never underestimate the power of yourdetermination. Remember that setbacks are temporary, but the lessons they impart will shape you for life.Success is not a destination but a continuous journey, one that requires unwavering commitment, resilience, and a willingness to learn from every experience, both triumphs and failures alike. With this mindset, the possibilities are limitless, and the path to greatness lies before you, waiting to be conquered.篇3Climbing the Mountain of SuccessAs I sit here writing this essay, I can't help but reflect on the long and winding path that led me to where I am today – a successful university student on the cusp of graduating. It's been a journey filled with ups and downs, triumphs and failures, but most importantly, invaluable lessons that have shaped me into the person I am.Looking back, it almost seems surreal how far I've come. If you had told me years ago that I would one day be here, I would have laughed in disbelief. You see, my academic journey was far from a smooth ride. In fact, it was littered with obstacles and setbacks that threatened to derail me at every turn.My story begins in high school, where I struggled to find my footing. I was a bright student, but I lacked direction and discipline. I would coast through classes, barely scraping by with mediocre grades. It wasn't until my junior year that reality hit me like a ton of bricks – my lackluster performance had left meill-prepared for the challenges of college admissions.That was my first real taste of failure, and it stung deeply. I remember the disappointment on my parents' faces when I received rejection letters from the universities I had applied to. It was a humbling experience that forced me to take a hard look at myself and the choices I had made.But failure, as painful as it was, became my greatest teacher. It ignited a fire within me, a burning desire to prove to myself and others that I was capable of so much more. I made a vow to turn my academic life around, no matter how daunting the task may seem.And so began my ascent up the mountain of success – a treacherous climb fraught with challenges and setbacks, but one that I was determined to conquer.I started by adopting a new mindset, one that embraced hard work, discipline, and perseverance. I became a diligent student, spending countless hours poring over textbooks andseeking help from teachers whenever I encountered difficulties. I developed effective study habits and time management skills, ensuring that every minute counted towards my growth.But the road was far from smooth. There were times when I felt overwhelmed, when the mountain seemed too steep to climb. I faced moments of self-doubt, questioning whether I had what it took to succeed. But in those darkest moments, I drew strength from the memory of my past failures, using them as fuel to push forward.And push forward I did, one step at a time, one obstacle at a time. I began to see the fruits of my labor as my grades steadily improved. Small victories turned into bigger ones, and before long, I found myself at the top of my class.Yet, the true test of my perseverance came when I faced the daunting challenge of college admissions once again. This time, armed with a stellar academic record and a newfound sense of determination, I approached the process with unwavering confidence.The acceptance letters started pouring in, each one a testament to the journey I had undertaken. It was a moment of pure elation, a validation that all the sacrifices and hard work had been worth it.Now, as I stand on the precipice of graduation, I can't help but feel a sense of immense pride and gratitude. Pride for having overcome the obstacles that once seemed insurmountable, and gratitude for the lessons that failure taught me along the way.Failure, as I've come to realize, is not the enemy; it's a necessary part of the journey towards success. It's a humbling reminder that growth and achievement rarely come easy, and that true greatness lies in our ability to bounce back from adversity, stronger and wiser than before.As I step into the next chapter of my life, I carry with me the lessons that this incredible journey has imparted. I know that there will be more mountains to climb, more challenges to face, and more failures to overcome. But I also know that I possess the resilience, determination, and grit to conquer whatever obstacles lie ahead.For those of you who may be experiencing your own setbacks and failures, I implore you not to lose hope. Embrace the lessons that failure offers, use them as fuel to propel you forward, and never lose sight of your dreams. The road may be long and arduous, but the view from the summit is worth every ounce of effort.Success is not a destination; it's a journey – one that requires perseverance, resilience, and an unwavering belief in oneself. And while the path may be riddled with failures, it's those very failures that pave the way for our greatest triumphs.So, take heart, my fellow travelers, and remember – every failure is an opportunity, every setback a chance to rise, and every obstacle a stepping stone towards the summit of success.。

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