Function of a Random Variable¶

Definition¶

Let $X$ be a random variable and let $g\colon\mathbb{R}\to\mathbb{R}$ be a Borel-measurable function, meaning that preimages of Borel sets are Borel sets. Continuous and piecewise continuous functions satisfy this condition. The function of a random variable determined by $g$ is the random variable $Y=g(X)$ $g (X)$ .

Interpretation¶

A function of a random variable is a new quantity computed from the value of $X$ . On each outcome $\omega$ it takes the value $g(X(\omega))$ , so $Y=g(X)$ is the composition of $X$ and $g$ .

Arrows from the values minus 2 to 2 of X, each with probability 1/5, to the values 0, 1 and 4 of Y = X squared, with the probabilities that collect at each value.

In the figure $X$ is uniform on $\{-2,-1,0,1,2\}$ , each value with probability $\tfrac{1}{5}$ , and $Y=X^2$ $X^{2}$ . The values $-1$ and $1$ are both sent to $1$ , and $-2$ and $2$ both to $4$ , so their probabilities add: $\mathbb{P}(Y=0)=\tfrac{1}{5}$ $\frac{1}{5}$ , $\mathbb{P}(Y=1)=\tfrac{2}{5}$ and $\mathbb{P}(Y=4)=\tfrac{2}{5}$ $\frac{2}{5}$ . $X$ is discrete uniform, but $Y$ is not. In general, for discrete $X$ ,

\mathbb{P}(Y=y)=\sum_{x\colon g(x)=y}\mathbb{P}(X=x).

For a discrete $X$ this is the whole recipe: list the values of $g$ , and for each value $y$ add the masses of all $x$ with $g(x)=y$ $y$ . If $g$ is one-to-one, no masses merge, and $Y$ has the same masses as $X$ , placed at new values. In general $\{Y\in B\}$ is the event $\{X\in C\}$ $C}$ , where $C$ is the set of $x$ with $g(x)\in B$ $B$ ; the condition on $g$ in the Definition guarantees that $\{X\in C\}$ is an event, so that it has a probability.

Indicators are functions of this kind: the indicator of $\{X>10\}$ is $g(X)$ with $g(x)=1$ for $x>10$ and $g(x)=0$ otherwise. When only the average of $g(X)$ is needed, LOTUS skips finding the distribution of $Y$ : in the figure $\mathbb{E}[Y]=0\cdot\tfrac{1}{5}+1\cdot\tfrac{2}{5}+4\cdot\tfrac{2}{5}=2$ $2$ .

A common slip is to apply $g$ to the probabilities instead of the values. The masses of $Y=X^2$ are not the squares of the masses of $X$ .

Key Results¶

If $X$ is a random variable and $g$ is Borel measurable, then $g(X)$ is again a random variable. For a discrete $X$ , any real-valued assignment on its finite or countable set of values gives a random variable.
The distribution of $g(X)$ may differ substantially from the distribution of $X$ .
LOTUS can compute expectations of $g(X)$ without first deriving the full distribution of $g(X)$ .

Examples¶

If $X$ is temperature in Celsius, then a linear function of $X$ gives temperature in Fahrenheit.
If $X$ is a return, then $X^2$ measures squared return magnitude.
If $X$ is a waiting time, then an indicator of whether it exceeds ten minutes is a function of $X$ .

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 3
Probability Theory for EOR · Week 4

Aliases: transformed random variable Keywords: random variable transformation Symbols: element of, omega, real numbers, sum

Last updated: 20 October 2026 03:02