What is expected value of a sample mean?

Question

Let $X_1,. . . , X_n$ be a random sample from a population with variance $σ^2$ and mean $μ$. Designate the corresponding sample mean by $\bar{X}$. Show that $\mathbb{E}(X_i\bar{X})=σ^2/n + µ^2, i = 1, . . . , n.$ Help please! Can't do anything.

Answer 1

For some $i \in \{ 1, \dots, n\}$, $$X_i\bar{X}=\dfrac{X_1X_i+X_2X_i+\cdots+X_{i-1}X_i+X_i^2+X_{i+1}X_i+\cdots+X_nX_i}{n}$$ Now for some $j \in \{1, \dots, n\}$, where $j \neq i$, we have $$\mathbb{E}[X_jX_i]=\mathbb{E}[X_j]\mathbb{E}[X_i]=\mu^2$$ due to independence (random sampling). Furthermore, $\mathbb{E}[X_i^2] = \sigma^2+\mu^2$. Hence, $$\mathbb{E}[X_1X_i+X_2X_i+\cdots+X_{i-1}X_i+X_i^2+X_{i+1}X_i+\cdots+X_nX_i]=(n-1)\mu^2+\sigma^2+\mu^2$$ which simplifies to $n\mu^2+\sigma^2$.

Now multiply by $\dfrac{1}{n}$.