Let X and Y be independent random variables, uniformly distributed on [0,1] . Let U=min{X,Y} and V=max{X,Y} .

Question

Let a=E[UV] and b=E[V]

1. Find a
2. Find b
3. Find Cov(U,V) . You can give either a numerical answer or a symbolic expression involving a and b . (There are many possible equivalent answers.)

Answer 1

𝖤𝑈=𝖤𝑈1{𝑋≤𝑌}+𝖤𝑈1{𝑋>𝑌}=∫10∫1𝑥𝑥𝑑𝑦𝑑𝑥+∫10∫𝑥0𝑦𝑑𝑦𝑑𝑥=13.
EV= 1-EU = 1/3
COV(U,V) = E(UV) - EUEV=1/36

Answer 2

a=E[UV] = 1/4