Probability distributions for discrete random variables

By the end of this lesson

By the end you can define a discrete random variable, write its probability distribution table, use the fact that the probabilities always total $1$ to find an unknown value, and read a probability such as $P(X \geq 2)$ from the table.

Definition

Discrete random variable

A random variable is a variable whose value is the numerical outcome of a random experiment.

A random variable is discrete when its possible values can be listed, one by one, with a gap between consecutive values — typically counting values such as $0, 1, 2, 3, \dots$ .

The probability distribution of a discrete random variable $X$ lists every possible value $x$ together with its probability $P(X = x)$ .

Capital and small letters. The capital $X$ is the random variable itself: the rule "count the heads", whose value is not known until the experiment is done. The small $x$ stands for one particular value it can take, such as $2$ . So $P(X = 2)$ reads "the probability that the number of heads is 2".

Turning an experiment into numbers

Random variables are how we attach numbers to outcomes so we can calculate with them.

Coin tosses are heads and tails, which is awkward. But "the number of heads in three tosses" is a number: $0$ , $1$ , $2$ or $3$ . Now we can average it, measure its spread, and ask questions like $P(X \geq 2)$ .

A continuous quantity such as the exact height of a student is not discrete, because its value cannot be listed one by one. Heights also need a different theory, which comes later in the course.

An example distribution in picture form: values

0

3

with probabilities

0.1

0.2

0.3

and

0.4

. The heights are probabilities, so they must total

1

; a bar chart makes it obvious when a distribution is complete and when something is missing.

The two rules every distribution obeys

For a discrete random variable $X$ with possible values $x_1, x_2, \dots, x_n$ :

Every probability is between $0$ and $1$ : $0 \leq P(X = x_i) \leq 1$ .
The probabilities over all possible values total exactly $1$ :

$\sum_{i} P(X = x_i) = 1$

The symbol $\Sigma$ (capital Greek "sigma") means "add up". So this says: add $P(X = x_i)$ over every value $x_1, x_2, \dots, x_n$ , and the total is $1$ .

The second rule is the workhorse. It is what lets you find an unknown constant in a distribution, and it is the check that you have not missed a value out.

Worked example: Example 1: from an experiment to a table

A fair coin is tossed $3$ times. Let $X$ be the number of heads. Write down the probability distribution of $X$ .

Have a go first?

Check 1 — building a table

CalculateQuestion 1 · Apply

Multiple choiceQuestion 2 · Recall

Answer the 2 questions above to continue.