Question

V = max{X, Y } and U = min{X, Y }. Let X and Y be independent random variables each having the uniform distribution on [0, 1]. Find the expectation E(U) and cov(U, V ).Let X and Y be independent random variables each having the uniform distribution on [0, 1]. Find the expectation E(U) and cov(U, V ).

Answer 1

• E(U) =
  `(1)/(3)`
• cov(U,V) =
  `(1)/(36)`

Explanation:

• E(U)=EU1{X≤Y}+EU1{X>Y}

=
`\int\limits^1_0\int\limits^1_x {x} \, dydx + \int\limits^1_0\int\limits^x_0 {y} \, dydx`

• cov (U,V) = E(UV) - E(U)E(V)

E(UV) = 1/4 , E(U) = 1/3, E(V) = 2/3

con(U,V) = 1/4 - (1/3 * 2/3) = 1/36