Poisson Distributions

Calculating Probabilities

The Problem

Let X be a random variable following a Poisson distribution. All Poisson distributions have just one parameter: average rate, λ (lambda). For this problem, let X have rate parameter λ = 7. An example of where such a distribution may arise is the following:

Statistics, Inc., makes candy. Unfortunately, they are not too good at it. Every week, an average of 7 pieces are defective. These defective pieces will not kill a person, but they will cause the person’s left index finger to turn scarlet. Let X be the number of defective pieces of candy manufactured in a specified week.

From the description, we can tell that X has a Poisson distribution with rate parameter λ = 7. We know this because X is the number of successes measured over time (or space).

For those who like pictures, here is a graphic of the probability mass function of X. Note that it is skewed positive (right), that its sample space is non-negative integers, and that there is no upper bound to its sample space. In other words, S = {0, 1, 2, 3, …}.

0.05 0.10 0.15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  …

Continuing the candy example, let us determine the probability that there are exactly 15 defective pieces of candy produced in the week. In symbols, calculate P[X = 15].

Your Answer

In the box below, please enter the value of P[X = 15], where X ~ Poisson(λ = 7), then click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

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© Ole J. Forsberg, Ph.D. 2024. All rights reserved.   .