Calculating Cumulative 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 = 16.67. 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 16.67 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 follows a Poisson distribution with rate parameter λ = 16.67. We know X follows a Poisson distribution because it 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, and that there is no upper bound to its sample space. In other words, S = {0, 1, 2, …}.

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Continuing the candy example, let us determine the probability that there is no more than 4 pieces of sour apple candy in the bag. In symbols, calculate P[X ≤ 4].

Your Answer

Another Poisson Distribution

Would you like to continue working on this topic? If so, click here for another Poisson distribution.

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