Question

at XX e random variables endh withy the Also, the correlation coeficients of Xi, x2, x3 are ρ12 =-0.3,P13 Find the mean and variance of 0.6, p23 = 0.5

Answer 1

$E(X) = \int _0^1\frac{3x^2}{2}\left(2-x\right)dx=\frac{5}{8}\quad \left(\mathrm{Decimal:\quad }\:0.625\right)$

$E(X^2) =\int _0^1\frac{3x^3}{2}\left(2-x\right)dx=\frac{9}{20}\quad \left(\mathrm{Decimal:\quad }\:0.45\right)$

Var(X) = E(X^2) - (E(X))^2

= 0.45 - 0.625^2

= 0.059375

E(Y) = E(X1 + 3 X2 + 2 X3) = E(X) + 3 E(X) + 2 E(X)

= 6 E(X) = 6 *5/8 = 3.75

Var(Y) = Var(X1 + 3 X2 + 2 X3)= Var(X) + 9 Var(X) + 4 Var(X) + 2 * 3 *$\rho_{12}$ Var(X) + 2 * 2 *$\rho_{13}$ Var(X) +2*$\rho_{23}$*2*3*Var(X)

= Var(X) (14 + 6 * $\rho_{12}$ + 4*$\rho_{13}$ + 12 $\rho_{23}$)

= 0.059375* ( 14 + 6 * (-0.3) + 4 * 0.6 + 12 * 0.5)

= 1.223125