Distributions

This file consists of descriptions, references, and examples for common distributions (both discrete and continuous).

Continuous Distributions

The Poisson distribution is the binomial distribution as the number of trials approaches infinity.

The exponential distribution is the geometric distribution as the number of trials approaches infinity. In its most common use case, it models the probability that a certain amount of time elapses before an event occurs, where the probability of the event occuring at one instance is independent of the probability of that event occuring at a prior instance.
It can be alternatively defined as modeling the time until 1 success in a Poisson process
A good description of the distribution as well the intuition behind it can be found here: https://www.probabilitycourse.com/chapter4/4_2_2_exponential.php

The Gamma distribution is a modification of the exponential distribution, where the rate of decay varies at different values of the r.v. It takes two inputs: shape (α) and rate (λ), where λ is the rate of decay and α is the rate of change of the rate of decay. If α = 1, the Gamma distribution is equivalent to the exponential distribution because the rate of decay λ does not vary. If α < 1, the Gamma distribution models the probability that a certain amount of time elapses before an event occurs, where the probability of the event occuring grows increasingly less likely for larger intervals of time. If α > 1, the Gamma distribution models the probability that a certain amount of time elapses before an event occurs, where the probability of the event occuring grows increasingly more likely up to a certain duration of time (mode) and then decays for greater durations of time.

The normal distribution models a r.v. that has a most likely value (mode) and decays for any deviation from that value. It looks like a symmetric "bell shape". It is also referred to as the "bell curve" or the "Gaussian" distribution.
Most random variables do not follow this distribution, but it is very relevant because the expectation of a randomly chosen sample approaches the normal distribution as the sample size increases. This is important practically because many random variables can be modeled as a sum of many other random variables and hence we can argue that their distribution is approximated by the normal distribution.
A good description of it and its motivations can be found here: https://www.probabilitycourse.com/chapter4/4_2_3_normal.php
A good description of its relevance in practical applications, using the CLT, can be found here (with pictures): https://towardsdatascience.com/the-central-limit-theorem-and-its-implications-4a7adac9d6de