# 3 Discrete Distributions

## 3.1 Uniform distributions

Consider the uniform distribution on \(\{1, \dots, n\}\).

Its mass function (when defined on the integers) is given by

Here is a graph of this function

Its distribution function (when defined on the reals) is given by

Here is a graph of this function

## 3.2 Binomial distributions

Consider the binomial distribution with the following parameters.

Here is a plot of its mass function

Here is a plot of its (cumulative) distribution function

## 3.3 Geometric distributions

Here is a plot of its mass function

Here is a plot of its (cumulative) distribution function

## 3.4 Poisson distributions

Here is a plot of its mass function

Here is a plot of its (cumulative) distribution function

```
curve(ppois(x, lambda), -1, 30, 1e3, lwd = 2, xlab = "", ylab = "")
abline(h = 0, lty = 2)
abline(h = 1, lty = 2)
```

Let’s check the validity of the Law of Large Numbers.