HW2-Sol

Problem 3.3.17

Problem 3.4.2

Problem 3.5.17

Problem 3.6.5

Problem 3.7.7

Problem 4.2.4

Problem 4.3.6

Problem 4.4.7

Problem 4.5.8

Problem 4.6.6

Problem 4.7.5

Additional Problem

[Poisson Arrival/Poisson process definition] (N(t) is arrival count at time t)

I. N(0) = 0

II. The process has independent increments

P(N(t + s) − N(s) = n) = 𝑃(𝑁(𝑡) = 𝑛)

III. The number of events in any interval of length t is Poisson distributed with mean λt.

That is for all s, t ≥ 0

P(N(t + s) − N(s) = n) =(𝜆𝑡)𝑛

𝑛!𝑒−𝜆𝑡 n = 0,1,2

For inter-arrival time τ, at some arrival time s

P(τ ≤ t) = 1 − 𝑃(𝜏 > 𝑡) = 1 − 𝑃(𝑁(𝑠 + 𝑡) − 𝑁(𝑠) = 0) (cause 𝜏 > 𝑡 no arrival at (s, s + t))

= 1 − 𝑃(𝑁(𝑡) − 𝑁(0) = 0) 𝑏𝑦 𝐼𝐼 independent increment by definition - *

= 1 − P(N(t) = 0) = 1 − 𝑒−𝜆𝑡 (is independent to s)

Inter-arrival time fallows exponential distribution