# Thread: [SOLVED] Quick question on working with cdfs to find pdfs

1. ## [SOLVED] Quick question on working with cdfs to find pdfs

Let $Y$ be a random variable and denote its pdf $f_Y$

In general, let $g$ be an increasing function on the range of $Y$. Define $W=g^{-1}(Y)$. Show that $f_W(w) = f_Y(g(w))\cdot g^{\prime}(w)$

My question is how to incorporate the fact that g is an increasing function on the range of Y into my proof. I believe it belongs when I mess with the inverse, but I'm not sure how to say how I use that fact correctly.

Work:
$F_W(w) = P(W

$f_W(w)=\frac{d}{dw}F_W(w)=\frac{d}{dw}F_Y(g(w))=f_ Y(g(w))\cdot g^{\prime}(w)$

2. $P(g^{-1}(Y)

This would not necessarily be true if $g$ were not an increasing function. Try taking the $cosine$ of both sides.

$\frac{d}{dw}F_W(w)=\frac{d}{dw}F_Y(g(w))$

Niether would this.