Symmetric Distribution¶

Definition¶

Let $X$ be a random variable and let $a\in\mathbb{R}$ $R$ . The distribution of $X$ is symmetric about $a$ if $X-a$ and $a-X$ have the same distribution.

Interpretation¶

A distribution is symmetric about $a$ when deviations below $a$ behave exactly like deviations above $a$ : $X$ is as likely to fall at least a distance $t$ below $a$ as at least a distance $t$ above it. The graph of the PMF or density is then a mirror image in the vertical line at $a$ .

A density symmetric about a point a, with equal shaded tails below a - t and above a + t.

The density in the figure is symmetric about $a$ , so $f_X(a-t)=f_X(a+t)$ for every $t$ , and the two shaded tails $\mathbb{P}(X\le a-t)$ and $\mathbb{P}(X\ge a+t)$ have the same area. This holds for every distance $t$ , not just the one drawn.

Pick a centre $a$ and a distance $t$ : the distribution is symmetric about $a$ when the two tails match for every $t$ . The exponential distribution fails for every $a$ .

In the Definition, $X-a$ is the deviation above $a$ and $a-X$ the deviation below it; symmetry says that the two have the same distribution. An equivalent condition is $\mathbb{P}(X\le a-t)=\mathbb{P}(X\ge a+t)$ for all $t$ . With $a=0$ and the standard normal distribution, this is the identity $\Phi(-z)=1-\Phi(z)$ $Φ (z)$ .

Symmetry fixes the centre. The point $a$ is a median: $\mathbb{P}(X\le a)=\mathbb{P}(X\ge a)$ $a)$ , and the two add up to at least $1$ , so each is at least $\tfrac{1}{2}$ . If the mean exists, it equals $a$ , since linearity gives $\mathbb{E}[X-a]=\mathbb{E}[a-X]=-\mathbb{E}[X-a]$ $a]$ , so $\mathbb{E}[X-a]=0$ $0$ .

The converse fails: mean and median can coincide for a distribution that is not symmetric. Symmetry is also not special to the normal distribution; the uniform distribution on $(a-c,a+c)$ and a fair die, symmetric about $3.5$ , are symmetric too.

Key Results¶

Symmetry about a centre means left and right deviations have the same distribution.
If the distribution is symmetric about $a$ , then $a$ is a median, and the mean equals $a$ whenever it exists.
A symmetric distribution need not be normal.

Examples¶

A normal distribution is symmetric about its mean.
A fair die distribution is symmetric around three and a half.
A two-point distribution with equal mass at negative one and one is symmetric about zero.

Prerequisites¶

Distribution

Sources¶

Probability Theory for EOR · Week 6

Aliases: symmetry of a random variable, symmetric random variable Keywords: distribution symmetric about a point Symbols: element of, phi, real numbers

Last updated: 20 October 2026 03:02