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Q. 9.1

Expert-verified

A First Course in Probability

Found in: Page 413

Book edition 9th

Author(s) Sheldon M. Ross

Pages 432 pages

ISBN 9780321794772

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Short Answer

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.?

The answer of each parts is

(a) $e^{- 6}$

(b) $6$

(c) $3.40 P . M .$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Part (a) Step 1: Given Information

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

Part (a) Step 2: Explanation

We are taking random variable $N (2)$ . So, the required probability is

$P (N (2) = 0) = e^{- 6}$ $N (2)$

Part (b) Step 1: Given Information

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

Part (b) Step 2: Explanation

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

$λ = 2 \cdot 3 = 6$

Part (c) Step 1: Given Information

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

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

$5 E (T) = \frac{5}{3}$

Hence, the time of fifth arrival after $2 P . M$ is $3 : 40 P . M .$

$λ = 3$

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