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Answers without the blur. Sign up and see all textbooks for free! Q. 9.1

Expert-verified Found in: Page 413 ### A First Course in Probability

Book edition 9th
Author(s) Sheldon M. Ross
Pages 432 pages
ISBN 9780321794772 # 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) $3.40P.M.$

See the step by step solution

## 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 $N\left(2\right)$. So, the required probability is

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

## 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 $\lambda =3$, so $N\left(2\right)$ has poisson distribution with rate $2$.

$\lambda =2·3\phantom{\rule{0ex}{0ex}}=6$

## 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 $\lambda =3.$Hence, the average time of five arrivals is given as

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

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

$\lambda =3$ ### Want to see more solutions like these? 