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Q. 9.1

Expert-verifiedFound in: Page 413

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) $3.40P.M.$

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

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

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

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

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 $2$.

$\lambda =2\xb73\phantom{\rule{0ex}{0ex}}=6$

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

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$3:40P.M.$

$\lambda =3$

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