You have two hours and fortyfive minutes to complete the exa

You have two hours and forty-five minutes to complete the exam. This is a Closed-book exam. However, you are allowed to use a 8.5 times 11 help sheet during the. exam. Show your work! A random variable X has the pdf given by fx(x) = 3x^2, 0 LE x LE 1 Find: The density function fy(y) where Y = square root X. The density function fy(y) where Y = X^2.

Solution

1. pdf of X is given as f_X(x)=3x²          0<x<1

a) Y=sqrt(X)

as 0<x<1   the range of y is 0<y<1

now Y=sqrt(X) so X=Y²    hence the jacobian is J=dX/dY=dY²/dY=2Y

hence the density function of Y=sqrt(X) is

f_Y(y)=3(y²)²*|J|=6y⁵         0<y<1       [answer]   [replacing X=Y²   in f_x(x)]

b) Y=X²       

as 0<x<1   the range of y is 0<y<1

now Y=X²   or X=sqrt(Y) hence the jacobian is J=dX/dY=dsqrt(Y)/dy=1/(2sqrt(Y))

hence the density function of Y=X² is

f_Y(y)=3(sqrt(y))²|J|=3*sqrt(y)/2       0<y<1    [answer]       [replacing X=sqrt(y) in f_x(x)   ]