Question

9. Let y= X2, where X has the pdf below (a) Find the mean of Y without finding the pdf of Y. (b) Write the pdf of Y:f<v). (c) Find the mean of Y using.ffy), confirming your answer in part (a).

Answer 1

a)

mean of Y =E(Y)=E(X²)=$\int_{0}^{\infty }$x² f(x) dx=$\int_{0}^{\infty }$x³e^-x dx=-x³e^-x-3x²e^-x-6xe^-x-6e^-x |^$\infty$0 =6

b)

here as pdf of Y =|dx/dy|f_x(g^-1(y))

for X=√Y

therefore |dx/dy| =1/(2√y))

hence pdf of Y f(y)= (1/(2√y)))*(√y*e^-√y) =(1/2)*e^-√y for 0 <y<$\infty$

c)

mean of Y =E(Y)=$\int_{0}^{\infty }$y*f(y) dy =$\int_{0}^{\infty }$(1/2)ye^-√y dy

letting √y=u

dy=2√y du =2 u du

hence E(Y)=$\int_{0}^{\infty }$u³e^-u du   =-u³e^-u-3u²e^-u-6ue^-u-6e^-u |^$\infty$₀ =6