3.2 The Law of the Unconscious Statistician

Recall that for a continuous rv, $Y$,

If $Y=g(X)$ then how do we know that this is equivalent to

A similar result is proved for discrete rvs in the notes. For simplicity of exposition, we assume here that the continuous rvs $X$ and $Y$ are non-negative.

We first turn the expectation of $Y$ into a double integral. Reversing the order of integration provides a term $𝖯\left(Y>t\right)$, which is equivalent to $𝖯\left(g(X)>t\right)$; returning to the original order of integration produces the required result. Along the way we will prove that (if $Y\ge 0$), $𝖤\left[Y\right]={\int }_0^{\mathrm{\infty }}{S}_Y(t)dt$, where $S_Y(y)$ is the survivor function of $Y$; this is of interest in its own right.

Unnumbered Figure: First link, Second Link

Using the left figure for the first change of order of integration and the right figure for the second change of order, and recalling that $y={\int }_0^{y}1dt$, we have:

as required.