THe number of calls received by a care towing service averages 16.8?

Question

per day (per 24 hour period) After finding the mean number of calls per hour find the probability that in a randomly selected hour the number of calls is 2

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 one hour.  X has the Poisson distribution with parameter λt = 0.7

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 = 16.8

t = 0.04166667 time unit(s)

this results in our random variable X ~ Poisson( 0.7 )

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Find P(X = 2 ) = 0.7 ^ 2 * exp( -0.7 ) / 2 ! = 0.1216634