Question: Pstat 120B - Dr. RavatHomework 2(a) Find the probability density function of U1=2x-1.(b) Find the probability density function of U2=x2.Use the method of inverse transformations to find the densities in parts ( c-d ) and compare youranswers to parts ( a-b ).(c) Find the probability density function of U1=2x-1. Compare to part (a).(d) Find the

Pstat $120$B $-$ Dr$.$ Ravat

Homework $2$

$($a$)$ Find the probability density function of $U_1=2x-1.$

$($b$)$ Find the probability density function of $U_2=x^2.$

Use the method of inverse transformations to find the densities in parts $(c-d)$ and compare your

answers to parts $(a-b).$

$($c$)$ Find the probability density function of $U_1=2x-1.$ Compare to part $($a$).$

$($d$)$ Find the probability density function of $U_2=x^2.$ Compare to part $($b$).$

$($e$)$ Compute $E(U_1),E(U_2)$ using the derived densities $f_{U_1}(u)$ and $f_{U_2}(u).$

$($f$)$ Find $E(U_1),E(U_2)$ by first finding the moments of $x$ and then using the properties of

expectation to find $E(U_1),E(U_2).$ Does your answer match to the answer of the previous

part?