$f(y)=\begin{cases} \frac{b}{y^2}, & y\ge b,\\ 0, & \mbox{elsewhere}\end{cases}$.
is a bona fide probability density function for a random variable, $Y$. Assuming $b$ is a known
constant and $U$ has a uniform distribution on the interval $(0, 1)$, transform $U$ to obtain a random variable with the same distribution as $Y$.
I have no clue how to get started on this question. Could anyone helps me get started on this question or give some hints?
Best Answer
The target distribution is characterized by the fact that any random variable $X$ with this distribution is such that, for every $x\geqslant b$, $$ P(X\geqslant x)=\int_x^\infty f=\int_x^\infty \frac{b}{y^2}\,\mathrm dy=\frac{b}x. $$ On the other hand, if $Y=\dfrac{b}U$ with $U$ uniform on $(0,1)$, then for every $y\geqslant b$, $$ P(Y\geqslant y)=P\left(U\leqslant \frac{b}y\right)=\frac{b}y. $$ Ergo.