2.6d Derived Distributions

Transformation Theorem

The Transformation Theorem in probability provides a systematic way to determine the distribution of a transformed random variable subject to some measurable function $g:\mathbb{R}\to \mathbb{R}$. More particularly,

• If $X$ is a discrete random variable with a probability mass function $p_X(x)$, then the PMF of $Y=g(X)$ is obtained by summing over all values of $x$ that map to $y$:
  $p_Y(y)={\sum }_{\{x:g(x)=y\}}^{}{p}_X(x),$
  which simplifies to $p_Y(y)=p_X(g^{-1}(y))$ if $g$ is invertible/bijective.
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• If $X$ is a continuous random variable with a probability density function $f_X(x)$, then the PDF of $Y=g(X)$ is obtained by