The lengths of the minor and major axes are used to summarize dust particles that are approximately elliptical in shape. Let $X$ and $Y$ denote the lengths of the minor and major axes (in micrometers), respectively. Suppose that $f_{X}(x)=\exp (-x), 0<x$ and the conditional distribution $f_{Y \mid x}(y)=\exp [-(y-x)], x<y .$ Answer or determine the following:
(a) That $f_{Y \mid x}(y)$ is a probability density function for any value of $x$
(b) $P(X<Y)$ and comment on the magnitudes of $X$ and $Y$.
(c) Joint probability density function $f_{X Y}(x, y)$.
(d) Conditional probability density function of $X$ given $Y=y$.
(e) $P(Y<2 \mid X=1)$
(f) $E(Y \mid X=1)$
(g) $P(X<1, Y<1)$
(h) $P(Y<2)$
(i) $c$ such that $P(Y<c)=0.9$
(j) Are $X$ and $Y$ independent?