Question

Please solve 6.66. Thank you in advanced.

6-66 Letxand y be independent random variables with variances o2 and o2,respectively. Consider the sum z=ax + (1-a)y 0-as l Find a that minimizes the variance of z.

Answer 1

for X and Y are independent

therefore Var(Z) =a²Var(X)+(1-a)² *Var(Y)

=a²$\sigma$₁²+(1-a)²*$\sigma$₂²

for above to minimum: (d/da)Var(Z) =0

(d/da)*(a²$\sigma$₁²+(1-a)²*$\sigma$₂²) =2a$\sigma$₁²-2(1-a)*$\sigma$₂² =0

(1-a)/a =$\sigma$₁²/$\sigma$₂²

1/a-1 =$\sigma$₁²/$\sigma$₂²

a =$\sigma$₂²/($\sigma$₁²+$\sigma$₂²)