Question

3. Suppose that X1, X2, , Xn are independent random variables with the same expectation μ and the same variance σ2. Let X--ΣΑι Xi. Find the expectation and variance of

Answer 1

Let X₁, X₂ , ... , X_n are independent random variables.

Also it is given as

E( X_i ) = $\mu$

and V( X_i ) = $\sigma ^2$

$E(\bar X)=E\left ( \frac{1}{n} \sum_{i=1}^{n}X_{i}\right )=\frac{1}{n}\sum_{i=1}^{n}E(X_{i})=\frac{1}{n}\sum_{i=1}^{n}\mu = \frac{n\mu }{n}=\mu$

Therefore expected value of $\bar X$ is equal to $\mu$

Let's find variance of $\bar X$

$V(\bar X)=V\left ( \frac{1}{n} \sum_{i=1}^{n}X_{i}\right )=\frac{1}{n^2}\sum_{i=1}^{n}V(X_{i})$

The covariance terms are zeros, because the given random variables are independents.

$V(\bar X)=\frac{1}{n^2}\sum_{i=1}^{n}\sigma^{2} = \frac{n\sigma^{2} }{n^2}=\frac{\sigma ^{2}}{n}$

Therefore variance of $\bar X$ is   $\frac{\sigma ^{2}}{n}$