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PSTAT 160B

Lecture 2

1.2 Arrival and Interarrival Times:

Let (Nt)t≥0 denote a Poisson process with parameter λ > 0.

Let X1 denote the time of the first arrival, X2 the waiting time between the first and the second arrival, X3 the waiting time between the second and third arrival, and so on...

Can we say something about the distribution of X1,X2,...?

Are they independent?

 Definition 1.7 Let X1,X2,... be a sequence of i.i.d exponential random variables with parameter λ > 0. For t > 0, let Nt = max{n ≥ 1 : X1 + ... + Xn ≤ t} with N0 = 0. Then (Nt)t≥0 defines a Poisson process with parameter λ > 0. Let Sn = X1 + ... + Xn for n = 1,2,... We call S1,S2,... the arrival times of the process, where Sk is the time of the k-th arrival. Furthermore, Xk = Sk − Sk−1 for k = 1,2,... is the interarrival time between the (k − 1)-th and k-th arrival, with S0 = 0. Fact Definitions 1.2 and 1.7 of a Poisson process are mathematically equivalent!

Poisson Process = counting process for which interarrival times are independent and identically distributed exponential random variables

Important Properties of exponential distribution

Recall from PSTAT 120A:

A random variable X is memoryless if, for all s,t > 0 we have

P[X > s + t |X > s] = P[X > t].

 Fact The exponential distribution is the only continuous distribution which is memoryless.

Minimum of independent exponential random variables:

 Proposition 1.8 Let X1,...,Xn be independent exponential random variables with parameters λ1,...,λn. Let M = min{X1,...,Xn}. (a) For t > 0 we have P[M > t] = e−t(λ1+...+λn). That is, M has exponential distribution with parameter λ1 + ... + λn. (b) For k = 1,...,n we have λk P[M = Xk] = . λ1 + ... + λn

Proof: See Lecture 2 Part 3.

Sum of i.i.d. exponential distributed random variables is gamma distributed.

 Proposition 1.9 For n = 1,2,... let Sn be the time of the n-th arrival in a Poisson process with parameter λ. Then Sn has a gamma distribution with parameters n and λ. The density function of Sn is given by λntn−1e−λt fSn(t) = for t > 0. (n − 1)! Mean and variance are n n E[Sn] = and Var(Sn) = 2. λ λ

Proof: See Assignment 1.

 Example 1.10 The Transit Center in Downtown Santa Barbara services three lines, 24X, 12X, and 20. The buses on each line arrive at the Transit Center according to three independent Poisson processes. On average, there is the 24X every 10 minutes, the 12X every 15 minutes, and the line 20 every 20 minutes. (a) When you arrive at the Transit Center what is the probability that the first bus that arrives is the 12X? (b) How long will you wait, on average, before some bus arrives? (c) You have been waiting 20 minutes for the 24X and have watched three line 20 buses go by. What is the expected additional time you will wait for your bus 24X?
 Example 1.11 The times when goals are scored in hockey are modeled as a Poisson process in a work by Morrison (1976). For such a process, assume that the average time between goals is 15 minutes. (a) In a 60-minute game, find the probability that a fourth goal occurs in the last 5 minutes of the game. (b) Assume that at least three goals are scored in a game. What is the mean time of the third goal?
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