Discrete Random Variable¶

Definition¶

A real-valued random variable $X$ is discrete if there is a finite or countably infinite set $D$ such that $\mathbb{P}(X\in D)=1$$1$. Its distribution is determined by the masses $p_X(x)=\mathbb{P}(X=x)$ on this set.

Interpretation¶

A discrete random variable puts all its probability on a list of values, finitely many or countably many $x_1,x_2,\ldots$. The probability of any statement about $X$ is found by adding the masses of the values for which the statement is true.

Here $X$ is the number of heads in two fair tosses. The labels under the bars show which outcomes give each value: $\{TT\}$ gives $0$, $\{HT,TH\}$ gives $1$ and $\{HH\}$ gives $2$, so the masses are $\tfrac{1}{4}$, $\tfrac{1}{2}$ and $\tfrac{1}{4}$, and they sum to $1$. A probability such as $\mathbb{P}(X\ge1)=\tfrac{1}{2}+\tfrac{1}{4}=\tfrac{3}{4}$ is a sum of bars, as in the second Key Result.

The list of values may be infinite as long as it is countable. The number of tosses of a fair coin up to and including the first head takes the values $1,2,3,\ldots$ with masses $\tfrac{1}{2},\tfrac{1}{4},\tfrac{1}{8},\ldots$, which sum to $1$ as a geometric series (first success distribution). Discrete does not mean integer-valued: a variable that takes only the values $0.5$ and $2.5$ is discrete too.

The PMF $p_X$ stores the whole distribution, and the CDF is a staircase that jumps by $p_X(x)$ at each value $x$. A quantity spread over a whole interval, such as an exactly measured waiting time, where every single value has probability zero, is not discrete; such variables are treated later as continuous random variables.

Key Results¶

• There is a countable set $D$ with $\mathbb{P}(X\in D)=1$$1$.
• For any set $A$, $\displaystyle \mathbb{P}(X\in A)=\sum_{x\in A\cap D}p_X(x)$$\sum _{x\in A\cap D}{p}_X(x)$.

Examples¶

• A fair die has values $1,\ldots,6$, each with probability $1/6$.
• A Poisson count has possible values $0,1,2,\ldots$, so a discrete support need not be finite.

Prerequisites¶

• Random Variable
• Countable and Uncountable Sets

Related Concepts¶

• Counting
• Support of a Random Variable
• Expectation
• Continuous Random Variable

Sources¶

• Probability Theory for EOR · Week 3
• Nicky van Foreest · Probability Distributions · Refresher

Aliases: discrete r.v., discrete rv Symbols: element of, intersection, sum

Last updated: 20 October 2026 03:02