Discrete Distributions

Introduction

In the previous topic, we looked at the concept of a probability mass function, but there are many different forms this can take depending on the nature of the problem you’re dealing with.

Geometric distribution

The probability of $k$ trials required before one success. Where the $k^{th}$ trial is a success.

$\Bbb{P}(X = k) = p(1-p)^{k-1}$

Where $p$ is the probability of success in each trail.

Binomial distribution

The probability of $k$ successes in $n$ trials.

$\Bbb{P}(X = k) = \dbinom{n}{k}p^k(1-p)^{n-k}$

Where $p$ is the probability of success in each trail.

Negative binomial distribution

The probability of $k$ trails being required to obtain $n$ successes.

$\Bbb{P}(X = k) = \dbinom{k - 1}{n - 1} p^n(1 - p)^{k-n}$

Where $p$ is the probability of success in each trail.

Poisson distribution

The Poisson distribution is an approximation of the binomial distribution, making it very similar. The key difference with Poisson is that it can take into account a rate of some kind.

$\Bbb{P}(X = k) = \dfrac{λ^k}{k!}e^{-λ}$

Where the rate is given by:

$λ = np$

With $n$ being the number of trials and $p$ being the probability of each trial being a success. Some questions make you model this as the rate in $T$ time units. In other words, two trains arriving per hour would have a rate of $2T$ where $T$ is the number of hours.

Some properties of Poisson distributions include:

$\begin{aligned} \Bbb{E}[X] &= λ \\\\ \Bbb{E}[X^2] &= λ^2 + λ \\\\ Var[X] &= λ \\\\ \end{aligned}$

Hypergeometric distribution

The probability of $k$ blue balls being chosen from a bag of $N$ balls, where a total of $n$ balls are chosen and $K$ balls in the bag were blue.

$\Bbb{P}(X = k) = \dfrac{\dbinom{K}{k} \dbinom{N - K}{n - k}}{\dbinom{N}{n}}$

Where $p$ is the probability of success in each trail.

More

More on this can be found on page 447 of the textbook.