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|x (y) = exp[−(y − x)], x < y. Answer or determine the following:

(a) That f_Y|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_XY (x, y).

(d) Conditional probability density function of X given Y = y.

(e) P(Y < 2 | X = 1) 

(f) E(Y | 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?