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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 ).

Respuesta :

Answer:

  • E(U) = [tex]\frac{1}{3}[/tex]
  • cov(U,V) = [tex]\frac{1}{36}[/tex]

Step-by-step explanation:

  • E(U)=EU1{X≤Y}+EU1{X>Y}  

= [tex]\int\limits^1_0\int\limits^1_x {x} \, dydx + \int\limits^1_0\int\limits^x_0 {y} \, dydx[/tex]

  • 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

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