# The Poisson distribution

## Understanding The Poisson Distribution

• The Poisson Distribution is a type of discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space.
• These events must occur with a known constant mean rate and independently of the time since the last event.
• Its notation is X~P(λ) where λ is the expected number of events per interval and is the only parameter of the distribution.

## Properties of The Poisson Distribution

• The Poisson distribution is characterised by the fact that the mean is equal to the variance, i.e., E(X) = Var(X) = λ.
• The possible values of a Poisson random variable go from 0 to infinity.
• The Poisson distribution applies to events in an interval of time or space which is subdivided into a large number of sub-intervals such that the probability of an event occurring in each subinterval is a small constant, denoted as λ.

## Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)

• The Probability Mass Function of a Poisson distribution gives the probability of observing exactly k events in an interval. It can be calculated using the formula λ^k * e^(-λ)/k!.
• The Cumulative Distribution Function for a Poisson distribution is the sum of the probabilities of the outcomes up to and including the defined upper limit.

## Working with The Poisson Distribution

• The Poisson distribution can be used to model various real-world phenomena including network traffic, number of phone calls received by a call center, the number of decay events per unit for a radioactive source, etc.
• Poisson Distribution becomes a good approximation of Binomial Distribution when the number of trials is large, the probability of success is small, and you’re interested in the number of successes.
• As with all distributions, the best way to understand them is to look at lots of examples and do lots of exercises. Thinking about the real-life context can also be very helpful.

Remember, mastering these concepts requires repeated review and practice. Refer to your textbook and class notes for additional detail and work through as many practice problems as you can.