# Expectation Operator Rules, Covariance and Standard Error of the Mean

Yao Yao on September 10, 2014

## 1. Expectation Operator Rules

Here proves the last rule:

Suppose the joint pdf of $X$ and $Y$ is $j(x,y)$, then

If $X$ and $Y$ are independent, then by definition $j(x,y) = f(x)g(y)$ where $f$ and $g$ are the marginal PDFs for $X$ and $Y$. Then

## 2. Covariance Again

If $X$ and $Y$ are independent, $Cov(X, Y) = 0$

## 3. Standard Error of the Mean

If $X_1, X_2 , \ldots, X_n$ are n independent observations from a population that has a mean $\mu$ and standard deviation $\sigma$, $\bar{X} = \frac{1}{n} \sum_n {x_i}$ is itself a random variable, and satisfy

• $E[\bar{X}] = \mu$
• $Var(\bar{X}) = \frac{\sigma^2}{n}$

The standard error of the mean (SEM) is the standard deviation of the sample-mean’s estimate of a population mean, i.e.

## 4. Proof of $Var(\bar{X}) = \frac{\sigma^2}{n}$

### 4.1 Proof I

Suppose $T = (X_1 + X_2 + \cdots + X_n)$, then

### 4.2 Proof II

Suppose $T = (X_1 + X_2 + \cdots + X_n)$, then