Question

An average call center employee receives 15 complaints per hour.

(a) The employee has just hung up, find the probability that over the next 2 years hours, he receives 25 complaints?

b) During one week of work (5 days), the likelihood of not receiving a complaint within the first 5 minutes of the first complaint.

Answer 1

A) This is an example of a Poisson's process where the parameter is the no. of complaints per hour.

For two hours complaints per hour.

Let X be the random variable for the number of complaints received in the next two hours.

The probability density function of a Poisson's distribution is given by

For the above case

and

so

(answer)

B) This case is an example of an exponential distribution. We know that the time period between two occurrences in a Poisson's distribution follows an exponential distribution.

Hence the time period between two consecutive complaints also follows an exponential distribution with parameter

.

The probability density function of an exponential distribution

Now, the parameter = 15 complaints per hour.

Hence is 15 complaints in 60 minutes.

Hence   is 1/4 complaints per minute.

And x = 5 minutes

Hence the likelihood of receiving no complaints in the next 5 minutes after the first complaint

Here we don't take into account the time elapsed till the first complaint since the exponential distribution has a memoryless property.

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