Support of a Random Variable¶

Definition¶

The course uses $\{x\colon p_X(x)>0\}$ for a discrete support and a positive-density region for standard continuous models. The topological support, which depends only on the distribution, is

\operatorname{supp}(X)=\{x\colon{}\mathbb{P}(|X-x|<\varepsilon)>0\quad \text{ for every }\varepsilon>0\}.

For the usual continuous densities, this is the closure of the positive-density region. If a density has been changed on a set of zero length, for example at finitely many points, use the neighbourhood definition: such changes do not change the support.

Interpretation¶

The support is the set of values a random variable can actually take, as far as probability can tell. For a discrete random variable it is the set of values with positive mass, $\{x\colon p_X(x)>0\}$ $0}$ .

Two panels: probability bars at 0, 1, 2 and 3 with these points marked as the support, and an exponential density whose support is the half-line from 0 onwards.

The left panel shows $\operatorname{Bin}(3,\tfrac{1}{2})$ : the circled points $0,1,2,3$ form the support, the only values with positive mass. The right panel shows a continuous example, the exponential distribution with rate $1$ , whose density is $0$ to the left of $0$ and positive from $0$ on; its support is the half-line $[0,\infty)$ , marked in red.

The support tells where formulas apply: a mass formula such as $\binom{3}{k}/8$ is meant only for $k=0,1,2,3$ $0, 1, 2, 3$ , and sums over the values of $X$ only need the values in the support.

The topological definition asks that every small interval $(x-\varepsilon,x+\varepsilon)$ around $x$ has positive probability. For a discrete variable with finitely many values this gives the same set as the positive masses; it differs only in edge cases, such as adding the endpoints $0$ and $1$ for $\operatorname{Unif}(0,1)$ , as in the Examples. A value outside the support is not merely unlikely: a whole interval around it has probability zero, and the CDF is constant there.

Key Results¶

The course discrete support is $\{x\colon p_X(x)>0\}$ $0}$ .
The topological support consists of points whose every open neighbourhood has positive probability, including boundary and accumulation points.

Examples¶

For $\operatorname{Bern}(p)$ with $0<p<1$ $1$ , the support is $\{0,1\}$ .
For $\operatorname{Unif}(0,1)$ , a positive-density region is $(0,1)$ , while the topological support is $[0,1]$ .

Prerequisites¶

Sources¶

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

Aliases: support of a distribution Keywords: support Symbols: epsilon, infinity

Last updated: 22 October 2026 19:04