Question

Help! Suppose Y follows the distribution fY(y) = 2ye^(-y^2) with y > 0. Find pdf for random variable S where S = Y^2. Find pdf for random variable U where U = S/(1+S). Confirm that fU(u) has the properties of a probability density function. I finish first part, can someone help me for last two part?

Answer 1

1) Find pdf for S
\[S=Y^{2}=h(y)=s \rightarrow y=h^{-1}(s)=\sqrt{s} \rightarrow \frac{ d }{ dy }h^{-1}(s)=\frac{ 1 }{ 2}s^{-\frac{ 1 }{ 2 }}\]
\[fU(u)=fU[h^{-1}(s)]\frac{ d }{ ds }h^{-1}(s)=2\sqrt{s}e^{-\sqrt{s}^{2}} \frac{ 1 }{ 2 }s^{-\frac{ 1 }{ 2 }}\]
\[=e^{-s}\]

Answer 2

sorry is fS(s) rather fU(u) in above reply. is above correct?

Answer 3

can someone help me for last two part? (2) pdf for U = S/(1+S) and (3) Confirm that fU(u) has the properties of a probability density function