Number of calls that come into a mail order company follows a Poisson distribution.?

Question

All calls answered by single operator. Additional operator will be needed if calls exceed 20 per hour. Manager oserves that 9 calls came in during a randomly selected 15-minute period.

a) if calls are actually 20 per hour, what is probability that 9 or more will come in during a given 15-minute period?

b) If rate is 30 per hour, what is probability that 9 or more will come in during a 15-minute period?

c)Based on results from a and b, do you think that the rate of incoming calls is more likely to be 20 or 30 per hour?

d) would you hire another operator, why?

Answer 1

The Poisson distribution can be derived from the binomial distribution.  The Poisson is nothing more than the limiting case of the Binomial where n is large and p is small.

A good way to identify when you need to use the Poisson distribution is when the problem requires you to use a rate.  This is not always true, but more often than not remembering this will help you to identify a Poisson model.

Let X be the number of calls in 15 minues.  X has the Poisson distribution with parameter λt = 5

In general you have:

X ~ Poisson( λt )

P(X = x) = ( λt )^x * exp( -λt ) / x! for x = 0, 1, 2, 3, 4, ...

P(X = x) = 0 otherwise

the mean of the Poisson distribution is the parameter, λt

the variance of the Poisson distribution is the parameter, λt

In this problem we have

λ = 20 per hour

t = 0.25 time unit(s)

this results in our random variable X ~ Poisson( 5 )

Find P( X ≥ 9 ) =

∞

∑ P(X = i)

i= 9

This sum is an infinite sum and is not easy to solve so instead let's rewrite the sum in terms of a finite sum.

Find P( X ≥ 9 ) = 1 - P( X < 9 ) =

. . .  8

1 - ∑ P(X = i)

. . . i=0

= 0.06809363

if the rate was 30 per hour then

X ~ Poisson( 7.5 )

Find P( X ≥ 9 ) =

∞

∑ P(X = i)

i= 9

This sum is an infinite sum and is not easy to solve so instead let's rewrite the sum in terms of a finite sum.

Find P( X ≥ 9 ) = 1 - P( X < 9 ) =

. . .  8

1 - ∑ P(X = i)

. . . i=0

= 0.3380329

(c) and (d) are up to you.