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4.5 Key definitions and Relationships

Let $X$ be a rv and let $Y=g(X)$ where $g$ is a real-valued function.

1.

If $X$ is discrete then so is $Y$ , and $p_{Y}(y)=\sum_{x:g(x)=y}p_{X}(x)$ .
2.

Distribution function (cdf) method: $F_{Y}(y)=\operatorname{\mathsf{P}}\left({Y\leq y}\right)=\operatorname{\mathsf% {P}}\left({g(X)\leq y}\right)$ . The right hand side must be evaluated from knowledge of $X$ . If $X$ is continuous then differentiation gives $f_{Y}(y)$ .
3.

Density function (pdf) method: if $X$ is continuous and $g$ is 1-1 then $Y$ is continuous and $f_{Y}(y)=f_{X}(x)|dx/dy|$ , where the right hand side is evaluated at $x=g^{-1}(y)$ .
4.

Be careful to also specify the range of $Y$ .
5.

PIT: if a continuous rv, $V$ , has a cdf of $F$ then $F(V)\sim{\operatorname{\mathsf{Unif}}}(0,1)$ ; if $U\sim{\operatorname{\mathsf{Unif}}}(0,1)$ and $F$ is a cdf of a continuous rv then $F^{-1}(U)$ is a rv whose cdf is $F$ .