Two continuous r.v.'s X and Y have joint PDF given by fx,y (x, y) = e, 0≤x≤ 2y <∞. O Exponential(1/2) Uniform[0, 2y) Gamma(y, 2) Use the facts that fy (y) fx (x) = ffx,y (x, y) dy = e, x > 0, ½ Then the conditional distribution of X|Y=y is ? ON(0,1) r2y = f fx,y (x,y) dx, and x=

If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?

Suppose Y1,Y2,··· ,Yn are i.i.d. continuous uniform(0,1) distributed.(a) Prove that the kth-order statistic, Y(k), has Beta distribution with α = k and β = n−k+ 1.(b) What is the distribution of the median of Y1,Y2,··· ,Yn when n is an odd integer?(c) Find the joint density of the middle two of Y(1),Y(2),··· ,Y(n) when n is an even integer