What Is a Discrete Random Variable and Its Probability Distribution?

A discrete random variable assigns a numerical value to each outcome of a random experiment and can take only a finite or countably infinite set of values. Its probability distribution lists every possible value and the probability that the variable takes that value.

How It Works

A random variable is usually represented by a capital letter such as $X$. A particular possible value is represented by a lowercase letter such as $x$, and its probability is written $P(X=x)$.

“Discrete” means the possible values can be listed, even when the list is infinite. Counts such as $0,1,2,\ldots$ are discrete, whereas measurements such as time or height are usually continuous. The probabilities describe uncertainty before the experiment; after an outcome occurs, $X$ takes one specific value.

For a valid discrete probability distribution:

1. Every probability must satisfy $0 \leq P(X=x) \leq 1$.
2. The probabilities of all possible values must add to $1$:

$$\sum P(X=x)=1$$

For example, two fair coins are tossed and $X$ is the number of heads obtained. The equally likely outcomes are HH, HT, TH and TT.

The probabilities form a valid distribution because

$$\frac{1}{4}+\frac{1}{2}+\frac{1}{4}=1.$$

The distribution can also be used to calculate the expected value:

$$E(X)=\sum xP(X=x)=0\left(\frac{1}{4}\right)+1\left(\frac{1}{2}\right)+2\left(\frac{1}{4}\right)=1.$$

This means the long-run average number of heads per experiment is $1$, not that every experiment produces exactly one head.

A common misconception is that all values of a discrete random variable must be equally likely. They do not: only the probabilities must be valid and sum to $1$.

Exam Technique

This is IB Mathematics: Applications and Interpretation SL 4.7, common to SL and HL and testable in Paper 1 or Paper 2. A GDC may be used on every AI paper. In an exam, define $X$ clearly, show that the probabilities sum to $1$, and use $E(X)=\sum xP(X=x)$ when asked for the expected value or fairness of a game.