For each of the following find the pdf of Y For each of the

For each of the following, find the pdf of Y.

For each of the following, find the pdf of Y. Y=X^2 and fx(x) = 30x^2(1-x)^2, 0

We begin with the cumulative distribution function of Y :
1. FY (y) = P(Y y) = P(X² y) = P(X y^1/2)

So far we have
FY (y) = FX(y^1/2)

To find the pdf of Y we simply differentiate both sides wrt to y:
fY (y) = 1/2*y^-1/2*fx(y^1/2)
where, fX(·) is the pdf of X which is given.

So fY (y) = 1/2*y^-1/2*30y(1-y)=15*y^-1/2*y(1-y)

2. Similarly we will get fY (y) = 1/2*y^-1/2*fx(y^1/2)
where, fX(·) is the pdf of X which is given.

fY (y) = 1/2*y^-1/2

^3.FY (y) = P(Y y) = P(-In x y) = P(X y^-e)

To find the pdf of Y we simply differentiate both sides wrt to y:
fY (y) = -e x^-e-1*fx(y^-e)=-e x^-e-1*(n+m+1)!/n!m!*(y^-e)ⁿ*(1-y^-e)^m