Problem 143E

Refer to Exercise 3.142 (c). If the number of phone calls to the fire department, Y , in a day has a Poisson distribution with mean 5.3, what is the most likely number of phone calls to the fire department on any day?

Reference

Let p(y) denote the probability function associated with a Poisson random variable with mean λ.

c Notice that the result in part (a) implies that Poisson probabilities increase for a while as y increases and decrease thereafter. Show that p(y) maximized when y = the greatest integer less than or equal to λ.

Solution

Step 1 of 1

We have to find the no.of phone calls per day to the fire department

Let Y is the no.of phone calls to the fire department

Here

Here given that Y is following poisson distribution with mean 5.3

The mean of the poisson distribution is

We know that P(Y) is maximised when Y= greatest integer less than or equal to

Here

Then y should be less than and should be greatest integer

So y=5

Hence we can expect 5 phone calls per day to the fire department