The Poisson random variable is the number of successes achieved when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success.

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The Poisson random variable is defined as the number of occurrences of a rare event in a fixed amount of time or space, with an assumed constant rate of occurrence. It does not depend on a sample size or population size, and it assumes the events are independent and randomly distributed in time or space. The scenario described in the statement is a hypergeometric distribution, which does depend on sample size and population size, and it assumes sampling without replacement.

The Poisson Random Variable Is the Number of Successes Achieved