Many-to-One Transformations of Discrete Random Variables

Find the PMF and expected value of $Y = g(X)$ for a discrete random variable $X$ when $g$ is many-to-one. Apply the formula $P(Y=y) = \sum_{x:\,g(x)=y} P(X=x)$ and the law of the unconscious statistician $E[g(X)] = \sum_x g(x)P(X=x)$ .

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Tutorial

Introduction

For a one-to-one transformation $Y = g(X),$ each value of $Y$ corresponds to exactly one value of $X,$ so $P(Y=y) = P(X = g^{-1}(y)).$

A transformation is many-to-one when two or more values of $X$ map to the same value of $Y.$ In this case, $P(Y=y)$ must accumulate the probability from every $x$ that lands on $y$ :

P(Y = y) = \sum_{x \,:\, g(x) = y} P(X = x).

For example, suppose $X$ has PMF

\begin{array}{|c|c|c|c|}\hline x & -1 & 0 & 1 \\ \hline P(X=x) & 0.3 & 0.5 & 0.2 \\ \hline \end{array}

and let $Y = X^2.$ The function $x \mapsto x^2$ sends both $-1$ and $1$ to $1,$ so

\begin{align*} P(Y=0) &= P(X=0) = 0.5, \\ P(Y=1) &= P(X=-1) + P(X=1) = 0.3 + 0.2 = 0.5. \end{align*}

The PMF of $Y$ is therefore

\begin{array}{|c|c|c|}\hline y & 0 & 1 \\ \hline P(Y=y) & 0.5 & 0.5 \\ \hline \end{array}