随机过程第二章

f (t ) = λe − λ t
X1 follows an exponential distribution with parameter λ
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2.2 Properties of Poisson processes
For any s>0 and t>0, {X2>t|X1=s} ⇔{0 event in (s, s+t]|X1=s} P{X2>t|X1=s} = P{0 event in (s, s+t]|X1=s} = P{0 event in (s, s+t]} (independent-increment) = P{0 event in (0, t]} (stationary-increment) = P{N(t)=0}= e-λt P{ X2≤ t|X1= s} = 1- e-λt
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2.2 Properties of Poisson processes
λh1e − λh ...λhn e − λh e − λ (t − h −...h ) = e − λt (λ t ) n
1 n1 1 n
(According to Eq.2-1-1)
n! = (Depends on the total number of subscribers and their arriving time)
Let N(t) denote the number of subscribers, and Si denote the arrival time of the ith customer. The revenue generated by this customer in (0,t] is t-Si. Adding the revenues generated by all arrivals in (0,t] N (t ) ⎡ N (t ) ⎤ ∑ (t − Si ) , E ⎢ ∑ (t − S i )⎥ ⎣ i =1 ⎦ i =1
n! P{ti ≤ Si ≤ t+hi, i=1,…n|N(t) = n} = n h1 ...hn t P{ti ≤ S i ≤ ti + hi , i = 1...n N (t ) = n } n! = n h1....hn t
taking the limits as hi→ 0 for all i, we obtain n! f S1 ..... S n N ( t )(t1 ,..., t n n) = n t
Sn = ∑ X k ,
k =1
n
n = 1,2......
Proposition(命题） 2.1: A Poisson process N={N(t), t≥0} with rate λ, the interarrival time {Xn, n≥1}are independently and identically distributed random variables, each following an exponential distribution with parameter λ. ( f(t)=λe-λt , t>0, mean=1/λ )
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2.1 Definition
Definition 2: The counting process {N(t), t≥0} is said to be a Poisson process having arrival rate λ, λ>0, if (a) N(0) = 0 (b) The process has independent increments (c) The number of events in any interval of length t is Poisson distributed with mean λt. t≥0
n! f S1 .....S n N ( t )(t1 ,..., t n n) = n , t
Proof: P{ti ≤ Si ≤ t+hi, i=1,…n|N(t) = n}
0 < t1 < ... < t n < t
P{one event in (t i , t i + hi ], 1 ≤ i ≤ n, no events elsewhere in (0, t)} = P{N (t ) = n}
f (t ) = λe − λ t
each interarrival time {Xn, n≥1} follows an exponential distribution with parameter λ.
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2.2 Properties of Poisson processes
Example 1 Suppose that people immigrate into a territory at a Poisson rate λ=1 per day. (a) What is the expected time until the tenth immigrant arrives? (b) What is the probability that the elapsed time between the tenth and the eleventh arrival exceeds two days?
Chapter 2 Poisson processes
2.1 Definition 2.2 Properties of Poisson processes 2.3 Nonhomogeneous Poisson processes 2.4 Compound Poisson processes 2.5 Filtered Poisson processes
P{N (t ) = n} = e −λt
(λ t ) n , n!
n = 0 ,1,2 ,......
(Eq.2-1-1)
E[N(t)] = λt
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2.2 Properties of Poisson processes
Interarrival time distribution Let Sn denotes the epoch of the nth arrival of N and define S0=0. The interval time Xn= Sn-Sn-1, so
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2.2 Properties of Poisson processes
Proof: {X1>t} ⇔ {N(t)=0} P{X1>t} = P{N(t)=0} = e-λt P{ X1≤ t} = 1- e-λt differentiating the two sides of equation with respect to t, We obtain the density of distribution of X1:
(n − 1)!
Proof: {Sn ≤ t} ⇔ {N(t)≥n} P{Sn ≤ t}= P{N(t)≥n} =
∑
k =n
∞
e −λ t
(λ t ) k k!
differentiating the two sides of equation with respect to t :
f (t ) = λe −λ t
f (t ) = λe − λ t
X2 follows an exponential distribution with parameter λ, and X1 and X2 are independent
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2.2 Properties of Poisson processes
Similarly, we obtain: P{ Xn>t|Xn-1=s} = e-λt P{ Xn≤ t|Xn-1= s} = 1- e-λt We obtain the interarrival time distribution:
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2.1 Definition
Consider a counting process N={N(t), t≥0}, where N(t) denotes the number of arrivals in the interval (0,t]. Definition 1 A counting process N ={N(t), t ≥0} is a Poisson process with rate λ >0, if it possesses the following properties: (a) N(0) = 0, (b) It satisfies the stationary and independent increment properties (c) P{N(h)=1}=λh+o(h) (d) P{N(h) ≥ 2}=o(h)
(λt ) n −1 (n − 1)!
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2.2 Properties of Poisson processes
Past arrival times given – Joint density of past arrival times Order statistics Let Y1, Y2……Yn are n random variables, if we arrange these random variables from small to big, note Y(1) = y1 is the smallest in the sequence, Y(2) = y2 is the second smallest, …. Y(n) = yn is the biggest in the sequence. Y(1) < Y(2)……< Y(n) , Y(1)……Y(n) or y1……yn are the order statistics of Y1…Yn. Let f is density of distribution of Yi, if f follows the uniform density over (0,t), the joint density of {Y(i)} is :