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# All the Distributions You Need to Know

Binomial distributions appear in many real-world contexts. If a situation meets all of the four following criteria, chances are you’re looking at a binomial distribution:

• There are only two possible and mutually exclusive outcomes — for example, yes or no, customer or not, etc. (The bi in binomial.)
• There is a predefined, finite, and constant number of repeated experiments or trials.
• All trials/experiments are identical in that they are all conducted in the same fashion as the others, yet are independent in that one trial’s outcome does not affect others’.
• The probability of success is the same in each of the trials.

Consider, for example, a company that wants to predict the chance that a customer will purchase a product after being exposed to identical advertisements. After determining p and the maximum amount of ads the company would like to run (n), the company can then use a binomial distribution’s mass probability function to determine how many advertisements are worth running in their marketing campaign.

The Bernoulli Distribution is the Binomial Distribution with only one experiment. It happens in an experiment with only two outcomes, successfully with probability p and unsuccessfully with probability q = 1 – p.

The Bernoulli distribution really isn’t a distribution as it is a special case of the Binomial distribution, but it’s good jargon to understand.

A Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a constant mean rate, independently of the time since the last event. The controllable factor of each distribution is λ, which is the mean rate.

For instance, if you are keeping track of the number of emails you receive every day and notice you receive an average of 14 a day. If receiving an email does not affect the arrival times of future emails, then the number of emails you receive a day probably obeys the Poisson distribution. there are many other examples of these distributions in other scenarios. For example, the number of phone calls received by a call center per day or the number of decay events per second from a radioactive source have been shown to follow a Poisson distribution.

The distribution follows the probability mass function of: