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Chapter 9: Q. 9.1 (page 413)

Events occur according to a Poisson process with rate λ = 3 per hour. (a) What is the probability that no events occur between times 8 and 10 in the morning? (b) What is the expected value of the number of events that occur between times 8 and 10 in the morning? (c) What is the expected time of occurrence of the fifth event after 2 P.M.?

Short Answer

Expert verified

The answer of each parts is

(a) e-6

(b) 6

(c) 3.40P.M.

Step by step solution

01

Part (a) Step 1: Given Information

We need to find the probability that no events occur between times 8and 10in the morning.

02

Part (a) Step 2: Explanation

We are taking random variable N2. So, the required probability is

P(N(2)=0)=e-6N2

03

Part (b) Step 1: Given Information

We need to find the expected value of the number of events that occur between times 8and 10in the morning.

04

Part (b) Step 2: Explanation

The number of events expected is 6as we haveNis a poisson pocess with rate λ=3, so N(2)has poisson distribution with rate 2.

λ=2·3=6

05

Part (c) Step 1: Given Information

We need to find the expected time of occurrence of the fifth event after 2P.M.

06

Part (c) Step 2: Explanation

The times of inter-arrivals in possion have Exponential distribution with parameter λ=3.Hence, the average time of five arrivals is given as

5E(T)=53

Hence, the time of fifth arrival after 2P.Mis3:40P.M.

λ=3

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