View Full Version : Combining random variables, squared

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".:/

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

03-11-2010, 12:45 PM

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

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