Answer to Question #338100 in Statistics and Probability for shar

Question #338100

The joint PDF of the random variables f(x,y)=8xy,0<x<1, 0<y<x

f(x,y)=8xy,0<x<1, 0<y<x. Find the marginal probability density function of Y.


1
Expert's answer
2022-05-10T18:43:46-0400

At first, we check that R2f(x,y)dxdy=1\iint_{{\mathbb{R}}^2}f(x,y)dxdy=1. R2f(x,y)dxdy=010x8xydydx=014x3dx=x401=1\iint_{{\mathbb{R}}^2}f(x,y)dxdy=\int_0^1\int_{0}^x8xydydx=\int_0^14{x^3}dx=x^4|_0^1=1. The marginal probability density of YY is: fY(y)=Rf(x,y)dx=y18xydx=4yx2y1=4(yy3)f_Y(y)=\int_{\mathbb{R}}f(x,y)dx=\int_y^18xydx=4yx^2|_y^1=4(y-y^3).

Answer: the marginal probability density function of YY is: fY(y)=4(yy3).f_Y(y)=4(y-y^3).


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