(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

X1 and X2 are independent random variables. They both have the same mean (mue). Their variances are s1^2 and s2^2 respectively, where s1^2 and s2^2 are known constants. It is proposed to estimate mue by an estimator T of the form T=c1X1 + c2X2.

Show that T will be unbiased if c1 + c2=1

and find an expression for var(T) in terms of c1, s1^2 and s2^2.

(assuming c1+c2=1)

2. Relevant equations

3. The attempt at a solution

I showed that T will be unbiased if c1+c2=1

For the next part this is what i did:

var(T) = var(c1X1+c2X2)

var(c1X1+c2X2) = E[(c1X1+c2X2)^2] + {E[c1X1+c2X2]}^2

and then after expanding and simplifying, i got:

var(T) = 2(mue)^2(c1^2 + 2c1c2 + c2^2)

I can easily change c2 in terms of c1 but how do put in terms of s1^2 and s2^2 as this is what they are asking for??

Thank you

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# Weighted verage of two variables with minimal variance

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