Question

Let X and Y be independent random variables with means EX = 10 and EY = 5 and standard deviations σ_X = 2 and σ_Y = 1.

Find the second moment E(X + Y + 1)²

Answer 1

solution:

Given that

X and Y are independent random variables

E[X] = 10 , E[Y] = 5

= 2 , = 1

Then Var(X) = = 4 , Var(Y) = = 1

E[X^2] - E[X]^2 = 4 , E[Y^2]-E[Y]^2 = 1

E[X^2] = 4+(10^2) , E[Y^2] = 1+(5^2)

E[X^2] = 104 , E[Y^2] = 26

If X and Y are independent then E[XY] = E[X]*E[Y]

Now, The second moment

E[ (X+Y+1)^2] = E[ (X^2 + Y^2 + 1 + 2XY + 2Y + 2X) ] [(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca ]

= E[X^2] + E[Y^2] + E[1] + E[2XY] + E[2Y] + E[2X]

  = E[X^2] + E[Y^2] + 1 + 2*E[X]*E[Y] + 2E[Y] + 2E[X] [ E[kX] = kE[X] , E[k] = k ]

= 104 + 26 + 1 + (2 * 10 * 5) + (2 * 5) + (2 * 10)

= 261

E[ (X+Y+1)^2] = 261