Probability distributions

In a nutshell

Probability distributions are often used to describe the different probabilities for a certain event. The discrete uniform distribution is a specific example of a probability distribution that has many applications.

Definitions

Here are a list of definitions you should know relating to probability distributions.

NAME	DEFINITION
Random variable	A random variable is an event that has a random outcome. It is represented with an uppercase letter. A specific outcome is represented with a lowercase letter.
Sample space	The sample space of a random variable is the set of values that the variable can take.
Discrete	A random variable is discrete if it only takes specific values.

Representing probability distributions

The two main ways to represent a discrete probability distribution are a probability mass function and a table.

Example 1

The result of a fair six-sided die is represented with a random variable, $X$ . Represent the distribution of $X$ .

First, identify the outcomes and probabilities.

A fair die will land on the numbers $1,2,3,4,5,6$ with an equal probability of $\dfrac{1}{6}$ .

Represent the probabilities using a function:

$P(X=x)=\dfrac{1}{6},\space x\in \{1,2,3,4,5,6\}$

As a table:

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X=x)$	$\dfrac{1}{6}$	$\dfrac{1}{6}$	$\dfrac{1}{6}$	$\dfrac{1}{6}$	$\dfrac{1}{6}$	$\dfrac16$

Note: The notation $P(X=x)$ means "the probability that the random variable $X$ takes the specific outcome $x$ ". It is also correct to write $x=1,2,3,4,5,6$ instead of $x=\{1,2,3,4,5,6\}$ .

The discrete uniform distribution

The discrete uniform distribution is a specific distribution that only takes certain values and every single probability is the same. The random variable $X$ in the example above is an example of a discrete uniform distribution.

Probability distribution problems

Sum of probabilities

When solving problems involving a probability distribution, you may have to use the fact that probabilities always add up to $1$ . With the notation used, this can be written as:

$\boxed{\sum_{x}{P(X=x)}=1}$

Example 2

The probability mass function representing a random variable $X$ is given to be:

$P(X=x)=kx,\space x\in \{1,2,3\}$

Find the value of $k$ .

First, find out all of the probabilities:

$P(X=1)=k(1)=k\\P(X=2)=k(2)=2k\\P(X=3)=k(3)=3k$

Then, use the fact that the probabilities add up to $1$ to form an equation in $k$ :

$k+2k+3k=1\\6k=1\\\underline{k=\dfrac{1}{6}}$

Inequalities

You may be asked to find the probability that a random variable satisfies a given inequality. To approach these questions, think about which specific outcomes satisfy the inequality and add them up.

Example 3

Using the probability distribution in the above example, find $P(0.5\le X<3)$ .

Find out which outcomes satisfy the inequality.

The only values that satisfy $0.5\le X < 3$ are $X=1$ and $X=2$ .

Add up the probabilities:

$\begin{aligned}P(0.5\le X< 3)&=P(X=1)+P(X=2)\\&=k+2k\\&=3k\\&=3(\dfrac{1}{6})\\&=\dfrac{1}{2}\end{aligned}$

$\underline{P(0.5\le X < 3) = \dfrac{1}{2}}$