1/ Let X1 and X2 be two independent random variables each with probability density function f_Xi(xi) = e^-xi , for x_i > 0 for i = 1,2.

(a) Find the joint probability density function of X1 and X2.

(b) Find P(X1 > 1, X2 < 1).

(c) Find P(X1 + X2 < 2).

2/ Let X be a continuous variable with probability density function

f(x) = kx(1 - x)² with0 < x < 1, 0 otherwise.

(a) Find a value of k so that f(x) is a proper density.

(b) Find the cumulative distribution function of X.

(c) Find P(0.25 < X < 0.75 | X > 0.5).