Question

The Poisson probability formula is shown to the right, where X is the number of times the event occurs and λ is a parameter equal to the mean of X. This distribution is oflen used to model the frequency with which a specified event occurs duting a particular period of time. P(X=x)=e−λ x

λ x
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Suppose that a hospital keeps records of emergency room traffic. These records reveal that the number of patients who arrive between 2 . P. M, and 3 P. M, has a Poisson distribution with parameter λ=5,2. Complete parts a through c. a. Determine the probability that, on a given day, the number of patients who arrive at the emergency room between 2 P.M. and 3 P.M, is exactly 4. P(X=4)=

Answer 1

P(X=4) = 0.168

Explanation:

Recall the formula for a Poisson probability distribution is
`\displaystyle P(X=x)=(e^(-\lambda)\lambda^x)/(x!)` where λ is the mean number of occurrences in the given interval and x is the desired number of outcomes in the given interval.

Given the parameter
`\lambda=5.2` and
`x=4`, then:

`\displaystyle P(X=4)=(e^(-5.2)\cdot5.2^4)/(4!)\approx0.168`

This means that the probability that, on a given day, the number of patients who arrive at the emergency room between 2 P.M. and 3 P.M is exactly 4 would be about 0.168