The **Poisson Distribution** is a discrete distribution named after French mathematician Simeon-Denis Poisson.

Unlike the Binomial Distribution that has only two possible outcomes as a success or fail, this distribution focuses on the number of discrete occurrences over a defined interval. It is used to estimate the probabilities of events that
occur randomly *per* some unit of measure.

This formula describes rare events and is referred to as the law of improbable events. The formula shown below calculates the probability of occurrences over an interval.

It can provide an approximation to the Binomial Distribution if the number of samples (n) is large and probability of a success (p) is small.

**The Poisson distribution exhibits the following:**

**The Poisson Distribution may apply when studying the:**

Notice the word "*per*" in each of the above statements.

A **random variable** that has a Poisson distribution must have a probability, p, of occurrence that is proportional to the interval length. The number of occurrences in an interval must also be independent events.

Both the Poisson distribution and Exponential distribution are used to to model *rates* but the latter is used when the data type is continuous.

The Exponential distribution has a Poisson distribution when:

1) the event can occur more than 1 time

2) the time between two successive occurrences is exponentially distributed

3) the events are independent of previous occurrences

If the **random variable, x,** has an Exponential distribution then the reciprocal (1/x) has a Poisson distribution.

The Poisson Distribution is applied to model the number of events (counts) or occurrence per interval or given period (could be arrivals, defects, failures, eruptions, calls, etc.). This models discrete random variables.

The Exponential distribution is applied to model the *mean* time (such as waiting times) between occurrences, time is a continuous variable.

A new website has an average random hit rate of 2.9 unique visitors every 4 minutes. What is the probability of getting exactly 50 unique visitors every hour?

**Given information:**

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