Discrete distributions

Discrete refers to when our sample space is countable, such as in the cases of coin tosses or rolling dice. 

In discrete probability distributions, the sample space is  and . 

The following are the six different kinds of discrete distributions that we often encounter in probability theory:

• Bernoulli distribution
• Binomial distribution
• Geometric distribution
• Hypergeometric distribution
• Poisson distribution

Let's define them in order.

For the Bernoulli distribution, let's use the example of a coin toss, where our sample space is Ω = {H, T} (where H is heads and T is tails) and p ∈ [0, 1] (that is, 0 ≤ p ≤ 1). We denote the distribution as B(1, p), such that the following applies:

But now, let's suppose the coin is flipped n times, each with the aforementioned probability of p for the outcome being heads. Then, the binomial distribution, denoted as B(n, p), states the following:

Therefore, we have the following:

Generally, the binomial distribution is written as follows:

The geometric distribution does not keep any memory of past events and so is memory-less. Suppose we flip our coin again; this distribution does not give us any indication as to when we can expect a heads result or how long it will take. So, we write the probability of getting heads after getting tails k times as follows:

Let's say we have a bucket filled with balls of two colors—red and black (which we will denote as r and b, respectively). From the bucket, we have picked out n balls and we want to figure out the probability that k of the balls are black. For this, we use the hypergeometric distribution, which looks as follows:

The Poisson distribution is a bit different from the other distributions. It is used to model rare events that occur at a rate, λ. It is denoted as P(λ) and is written as follows:

This is true for all cases of .