Combining random variables, squared

Chub

New Member
#1
Been sitting with this for a while, and hoping wome of you can assist me. I'm given a pdf f(y) = 6y(1-y) for 0<y<1, and I want to calculate the pdf of W=Y^2. I got som tools to calculate the pdf of W=XY, if X and Y are independent, but are these tools valid in this case? And even if they are, I get the wrong result. Have been trying too search the web for a while, but quite impossible since I can't search the "^2".:/
 

BGM

TS Contributor
#2
if W = Y^2, then Y = √W, dy/dw = 1/(2√w)
So using standard technique of Jacobian transformation,
fW(w) = fY(√w)|dy/dw| = 6(√w)(1 - √w)/(2√w) = 3(1 - √w), 0 < w < 1
 

Martingale

TS Contributor
#3
if \(W = Y^2\), then \(Y = \sqrt{W}, \frac{dy}{dw} = \frac{1}{2\sqrt{w}}\)
So using standard technique of Jacobian transformation,
\(f_{W}(w) = f_{Y}(\sqrt{w})\left|\frac{dy}{dw}\right| = 6(\sqrt{w})(1 - \sqrt{w})\frac{1}{2\sqrt{w}} = 3(1 - \sqrt{w}), 0 < w < 1\)

We do have LaTeX now...you should try it :D