# Discrete Distributions

This lesson will provide an overview of discrete distributions in JAX.

## We'll cover the following

## Discrete distributions

There are some scenarios where we need to sample the distribution from discrete events like a coin toss result, the value of a dice roll, or the number of people.

## Probability Mass Function (PMF)

The concept of CDF is the same for both continuous and discrete distributions, though PDF is replaced by the **Probability Mass Function (PMF)**. Here we calculate the probability for a given sample directly instead of differentiation.

$p_X(x) = P(X = x)$

## Poisson distribution

Poisson distribution is used to sample **unlikely events** (events with low probability).

$f(k; \lambda)= \Pr(X{=}k)= \frac{\lambda^k e^{-\lambda}}{k!}$

Both expected value and mean will be the same:

$E[X] = Var[X] = \lambda$

We can sample a Poisson distribution through `poisson(key, <λ's value>, size) `

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