RadfordMathematics.com

Online Mathematics Book

Discrete Random Variables & Probability Distribution Functions (PDF)


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.

Definition: Discrete Random Variable


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 \(1\).

Example

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:

\(X:\) the number obtained when rolling a dice.
Then this is a discrete random variable since the sum of the probabilities of each of these possible outcomes is equal to \(1\), indeed: \[\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6} = 1 \]

Probability Distribution Function (PDF)


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.

Example

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:

\(X\): the number obtained when we pick a ball at random from the bag
and given that its probability distribution function is: \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{8x-x^2}{40}\] Answer each of the following:
  1. State the possible values that \(X\) can take.
  2. Calculate the probability of picking a ball with \(2\) on it.
  3. Calculate the probability of picking a ball with \(4\) on it.

Solution

  1. Given the balls are numbered either \(2\), \(4\) or \(6\), the discrete random variable, \(X\), can take-on either of those values.
    We write all the values \(X\) can take inside a set \(x\): \[x=\left \{2, \ 4, \ 6 \right \}\] We're given the probability distribution function for \(X\): \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{8x-x^2}{40}\]
  2. The probability of picking a ball with \(2\) on it equals to the probability of \(X\) being equal to \(2\), that's \(P\begin{pmatrix} X = 2 \end{pmatrix}\).
    To calculate \(P\begin{pmatrix} X = 2 \end{pmatrix}\) we "simply" replace \(x\) by \(2\) in the function that was given to us in the question and calculate: \[\begin{aligned} P\begin{pmatrix} X = 2 \end{pmatrix} & = \frac{8\times 2-2^2}{40} \\ & = \frac{16-4}{40} \\ P\begin{pmatrix} X = 2 \end{pmatrix} & = \frac{12}{40} \end{aligned}\]
  3. The probability of picking a \(4\) is calculated in the same way, except we now replace \(x\) by \(4\): \[\begin{aligned} P\begin{pmatrix} X = 4 \end{pmatrix} & = \frac{8\times 4-4^2}{40} \\ & = \frac{32-16}{40} \\ P\begin{pmatrix} X = 4 \end{pmatrix} & = \frac{16}{40} \end{aligned}\] So the probability of picking a ball numbered \(4\) is \(\frac{16}{40}\).

Distribution Tables & Graphs

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:

  • a probability distribution table, or
  • a bar chart.
Each of these is illustrated in the example below.

Example: Distribution Table & Graph

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:

\(X\): the number obtained when we pick a ball from the bag.

The probability distribution function associated to the discrete random variable is: \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{8x-x^2}{40}\] We now learn how to draw a probability distribution table as well as its corresponding bar chart.

Probability Distribution Tables

To draw this discrete random variable's probability distribution table

  • top row: enter all the values \(x\) that the discrete random variable \(X\) can take.
  • bottom row: enter all of the corresponding probabilities, \(P\begin{pmatrix} X = x \end{pmatrix}\).

    Note: each of the probabilities in the second row is calculated by replacing the \(x\) in the function \(P\begin{pmatrix} X = x \end{pmatrix} = \frac{8x-x^2}{40}\) by the value of \(x\) directly above it in the table.
Doing so for our discrete random variable \(X\) leads to the following distribution table:

Probability Distribution Bar Graph

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.

Tutorial

Exercise

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: \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{8x-x^2}{40}\]

  1. Construct a probability distribution table for \(X\).
  2. Illustrate this probability distribution with a bar chart.
  3. Using your previous answers, state which value the discrete random variable \(X\) most likely to take?
  1. A discrete random variable \(X\) has a probability distribution function defined as: \[P\begin{pmatrix} X = x \end{pmatrix} = kx^2\] where: \(x = \left \{ 0, \ 1, \ 2, \ 3\right \}\).

    1. Find the value of \(k\).
    2. Calculate the probability that \(X = 2\).

  2. A discrete random variable \(X\) has a probability distribution function defined as: \[P \begin{pmatrix} X = x \end{pmatrix} = \frac{x}{k} \] where: \(x = \left \{ 1, \ 2, \ 3, \ 4, \ 5 \right \}\).
    1. Find the value of \(k\).
    2. Illustrate this discrete probability distribution in a table.

  3. A discrete random variable has a probability distribution function \(f(x)\), its distribution is shown in the following table:
    1. Find the value of \(k\) and draw the corresponding distribution table.
    2. Represent this distribution in a bar chart.
    3. Which value is the discrete random variable most likely to take?

Answers Without Working

    1. \(k = \frac{1}{14}\)
    2. \(P \begin{pmatrix} X = 2 \end{pmatrix} = \frac{2}{7}\) that's \(0.286\) (rounded to 3 significant figures).

    1. \(k = 15\)

    1. \(k=-0.1\)

      The probability distribution therefore becomes:
    2. The graphical representation, of this distribution, is shown in the following bar chart:

    3. The discrete random variable is most likely to take the value \(2\).