Show that $ Var[\sum_{i=1}^{n} a_{i} X_{i}] $ is minimized for $a_{i}$ = 1/n, i = 1,2,...n

Question

Let X1, X2, ..., Xn be a random sample from some density which has mean µ and variance $ σ^{2} $, if $ \sum_{i=1}^{n} a_{i} = 1$, show that $ Var[\sum_{i=1}^{n} a_{i} X_{i}] $ is minimized for $a_{i}$ = 1/n, i = 1,2,...n

So put the variance inside the summation $ Var[\sum_{i=1}^{n} a_{i} X_{i}] =\sum Var [a_{i}X_{i}] = \sum a_{i}^{2} \frac{n}{n-1} σ^{2} $

then differentiate against a

$ 2 \frac{n}{n-1} σ^{2}\sum a_{i} = 0 $ but $ \sum a_{i} = 1$

So what's wrong?

Answer 1

Since there is a constraint that $\sum_{i=1}^{n} a_{i} = 1$, one cannot use unconstrained optimisation like the OP did in the question details.

Hint to solve this optimisation problem: Use Lagrange constrained optimization technique.