Q. 9.1 - A First Course in Probability

Question

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.?

Step-by-Step Solution

Answer

The answer of each parts is

(a) $e^{-6}$

(b) $6$

(c) $3.40 P.M.$

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

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

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 $2$.

$$\begin{gathered} \lambda =2·3 \\ =6 \end{gathered}$$

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 $2 P.M$ is$3:40 P.M.$

$\lambda =3$