One-to-One Transformations of Discrete Random Variables

When we apply a function to a random variable, we get a new random variable. If the function is one-to-one (distinct inputs give distinct outputs), then the probabilities transfer directly through the inverse.

Let $X$ be a discrete random variable with PMF $p_X(x) = P(X = x),$ and let $g$ be a one-to-one function defined on the support of $X.$ Then $Y = g(X)$ is a discrete random variable with PMF

$$p_Y(y) = p_X\!\left(g^{-1}(y)\right)$$

for $y$ in the range of $g,$ and $p_Y(y) = 0$ otherwise.

This follows because $g$ is one-to-one: each value $y$ of $Y$ comes from exactly one value $x$ of $X,$ namely $x = g^{-1}(y).$ So the event $\{Y = y\}$ is the same as the event $\{X = g^{-1}(y)\},$ and they have the same probability.

For example, suppose $X$ has PMF

$$\begin{array}{c|ccc} x & 0 & 1 & 2 \\ \hline p_X(x) & 0.2 & 0.5 & 0.3 \end{array}$$

and let $Y = 3X + 1.$ The function $g(x) = 3x+1$ is one-to-one with inverse $g^{-1}(y) = (y-1)/3.$ The possible values of $Y$ are $1, 4, 7,$ and

$$p_Y(1) = p_X(0) = 0.2, \qquad p_Y(4) = p_X(1) = 0.5, \qquad p_Y(7) = p_X(2) = 0.3.$$

The probabilities are unchanged — they have just been relabeled.