Independent and identical

Question

I have $X_1,X_2, \cdots X_n$ that are iid from $N(\mu,\sigma)$. In my derivation of the expression

$E \big( \sum^n_{i=1} X_i^2 \big)$

I have written

$E \big( \sum^n_{i=1} X_i^2 \big) = \sum^n_{i=1} E \big( X_i^2 \big) = n E \big( X_i^2 \big) $

Is it true that first $=$ is because $X_1,X_2, \cdots X_n$ are independent and the second $=$ is because they are identical?

Answer 1

The expected value operator, $\mathbb{E}[.]$, is linear in the sense that

• $\mathbb{E}[X_1+X_2+\cdots+X_n]=\mathbb{E}[X_1]+\mathbb{E}[X_2]+\cdots+\mathbb{E}[X_n]$
• $\mathbb{E}[\lambda X]=\lambda\mathbb{E}[X] ,\quad \lambda\in \mathbb{R}.$

We have $$\mathbb{E}\left[\sum_{i=1}^{n}X_i^2\right]=\sum_{i=1}^{n}\mathbb{E}[X_i^2]$$ $X_1,X_2,\cdots,X_n$have same distribution, thus $$\mathbb{E}\left[\sum_{i=1}^{n}X_i^2\right]=\sum_{i=1}^{n}\mathbb{E}[X^2_i]=\sum_{i=1}^{n}(\sigma^2+\mu^2)=n(\sigma^2+\mu^2)$$