Suppose $X_1,….,X_n$ are $iid$ random variables and for each of them
$Variance(X_i)= \sigma^2$. $a_1…a_n$ are also real numbers and
$\sum_{i=1}^n a_i = 1$ If $S = \sum_{i=1}^na_iX_i$, prove $var(S)$ is
minimum if $a_i = 1/n, \ i=1,…,n$
So i did following:
$Var(S) = \sum_{i=1}^na_i^2Var(X_i) + 2\sum_{i<j}a_ia_jcov(X_i, X_j)$
I'm not sure how to take this from here. Any ideas?
Best Answer
Since $X_i$'s are independent, $\operatorname{Cov}(X_i,X_j)=0$ if $i\neq j$. It follows that $$\operatorname{Var}(S)=\operatorname{Var}\left(\sum_{i=1}^na_iX_i\right)=\sigma^2\sum_{i=1}^na_i^2.$$ We can bypass the lengthy Lagrangian method by using Cauchy-Schwarz inequality: $$1=\left(\sum_{i=1}^na_i\right)^2=\left(\sum_{i=1}^n\frac{a_i\sigma}{\sigma}\right)^2\leq\left(\sum_{i=1}^na_i^2\sigma^2\right)\left(\sum_{i=1}^n\frac{1}{\sigma^2}\right)=\frac{n\operatorname{Var}(S)}{\sigma^2}.$$ Now if $a_i=1/n$, then $$\operatorname{Var}(S)=\sigma^2\sum_{i=1}^n\frac{1}{n^2}=\frac{\sigma^2}{n},$$ which is exactly the lower bound given by Cauchy-Schwarz.