Question

Suppose that on the average a certain store serves 5 customers per hour. What is the probability that the store will serve 8 customers in a particular two-hour period?

Answer 1

The Poisson distribution :

Consider the number of times an event occurs ver an interval of time or space, and assume that

The probability of the event's occurence is the same for any two interval of time or space, and assume that

Whether the event occurs in any interval is independent of whether the event occurs in any other non overlapping interval.

Then, the probability that the event will occur x times in a specified interval is given as:

Let us consider that on the average certain store serves 5 customer per hour. Let us define the random variable X which represents the number of customers serves per hour follows a Possion distribution with mean . Thus, the Possion distribution formula is as below:

Now, we are intersted to compute the probability that the store will serve 8 cutomers in a particular two-hour period follows Possion distribution with mean

Thus, the Poisson distribution formula is as below:

We compute the probability that the store will serve 8 customers in a particular two-hour period by using the formula (2) as follows.

Now, we compute the probability P(X=8). We compute this probability as follows

  

  

Therefore, the probability that the store will serve 8 customers in a particular two-hours period is 0.8874