X1 and X2 are independently distributed random variables with
P(X1=Ɵ+1) = P(X1=Ɵ-1) = 1/2
P(X2=Ɵ-2) = P(X2=Ɵ+2) = 1/2

i)Find the values of a and b which minimize the variance of Y=aX1 + bX2 subject to the condition that E[Y]=Ɵ.
ii)What is the minimum value of this variance?

The answers at the back of the book says that a and b is 4/5 and 1/5 respectively and the variance is 4/5.
I don't know how they got that answer and I'm not even sure where to start...

Question

X1 and X2 are independently distributed random variables with
P(X1=Ɵ+1) = P(X1=Ɵ-1) = 1/2
P(X2=Ɵ-2) = P(X2=Ɵ+2) = 1/2

i)Find the values of a and b which minimize the variance of Y=aX1 + bX2 subject to the condition that E[Y]=Ɵ.
ii)What is the minimum value of this variance?



The answers at the back of the book says that a and b is 4/5 and 1/5 respectively and the variance is 4/5.
I don't know how they got that answer and I'm not even sure where to start...

anonymous · Answer

What i have found is 
a+b=1
E[Y]=E[X1]=E[X2]
and var(X1)=1 and var(X2)=4 which could be potentially incorrect.