In this section we learn about discrete random variables and probability distribution functions, which allow us to calculate the probabilities associated to a discrete random variable.
We start by defining discrete random variables and then define their probability distribution functions (pdf) and learn how they are used to calculate probabilities.
A discrete variable is a variable that can "only" take-on a finite number of values. For instance the number we obtain , when rolling a dice is a discrete variable as it can only equal to \(1, \ 2, \ 3, \ 4, \ 5, \) or \(6\).
A discrete variable is a discrete random variable if the sum of the probabilities of each of its possible values is equal to \(1\).
When we roll a single dice, the possible outcomes are:
\[1, \ 2, \ 3, \ 4, \ 5, \ 6\]
The probability of each of these outcomes is \(\frac{1}{6}\).
If we define the discrete variable \(X\) as:
Given a discrete random variable, \(X\), its probability distribution function, \(f(x)\), is a function that allows us to calculate the probability that \(X=x\).
In other words, \(f(x)\) is a probability calculator with which we can calculate the probability of each possible outcome (value) of \(X\).
\[f(x) = P\begin{pmatrix}X = x \end{pmatrix}\]
Note: we usually don't write \(f(x)\), we'll simply write \(P\begin{pmatrix} X = x \end{pmatrix}\) as it highlights the fact that we're dealing with probabilities.
A bag contains several balls numbered either: \(2\), \(4\) or \(6\) with only one number on each ball. A simple experiment consists of picking a ball, at random, out of the bag and looking at the number written on the ball.
Defining the discrete random variable \(X\) as:
To illustrate the probabilities of each of the possible values a discrete random variable \(X\) can take, it will often be useful to showcase all the possible values of \(X\) alongside the corresponding probability.
This is usually done in either:
We'll stick to the example we saw further up:
A game of chance in which we pick, at random, a ball from a bag. Each ball is numbered either \(2\), \(4\) or \(6\). The discrete random variable is defined as:
To draw this discrete random variable's probability distribution table
Using the probability distribution table we have above, we can illustrate this probability distribution in a bar chart.
For each of the possible values \(x\) of the discrete random variable \(X\), we draw a bar whose height is equal to the probability \(P\begin{pmatrix} X = x \end{pmatrix}\).
For the probability distribution we have above this would look like:
Looking at this graph allows us to determine, at a quick glance, which value \(X\) is most likely to take on.
Here we can say that there is a greater chance that \(X = 4\). In other words a ball picked at random from the bag is more likely to be numbered \(4\) than any other value.
A discrete random variable \(X\) can take either of the values: \[x = \left \{ 2, \ 4, \ 6 \right \}\] and has a probability distribution function (pdf) defined as: \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{8x-x^2}{40}\]