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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.

Tutorial

In the following tutorial, we learn more about what discrete random variables and probability distribution functions are and how to use them. Watch it before carrying-on.

Definition: Discrete Random Variable


Discrete Variables

A discrete variable is a variable that can "only" take-on certain numbers on the number line.
We usually refer to discrete variables with capital letters: \[X, \ Y, \ Z, \ \dots \] A typical example would be a variable that can only be an integer, or a variable that can only by a positive whole number.

Discrete variables can either take-on an infinite number of values or they can be limited to a finite number of values.

For instance the number we obtain , when rolling a dice is a discrete variable, which is limited to a finite number of values:\(1, \ 2, \ 3, \ 4, \ 5, \) or \(6\).

An example of a discrete variable that can take-on an "infinite" number of values could be: the number of rain drops that fall over a square kilometer in Sweden on November 25th.
Note: although this quantity can technically not be infinite, it is common practice and acceptable to assume so.

Discrete Random Variables

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 \(X\). \[P\begin{pmatrix}X = x \end{pmatrix} = f(x) \]

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 following tutorial and in the detailed example below.

Example: Distribution Table & Graph

We'll stick to the example we saw further up:

A game of chance consists of picking, 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: \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{8x-x^2}{40}\]
  1. Construct a probability distribution table to illustrate this distribution.
  2. Draw a bar chart to illustrate this probability distribution.
  3. Use the distribution table and bar chart to determine which value the discrete random variable \(X\) is most likely to take.

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

In the following tutorial we learn how to construct probability distributions tables and their corresponding bar charts. Make sure to watch before working through the exercises below.

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 \(2\).