Expectation Operator Rules, Covariance and Standard Error of the Mean

1. Expectation Operator RulesPermalink

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 AgainPermalink

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

3. Standard Error of the MeanPermalink

If $X_1,X_2,\dots ,X_n$ are n independent observations from a population that has a mean $\mu$ and standard deviation $\sigma$, $\overline{X}=\frac{1}{n}\sum _n{x}_i$ is itself a random variable, and satisfy

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

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 $Var(\overline{X})=\frac{{\sigma }^2}{n}$Permalink

4.1 Proof IPermalink

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

4.2 Proof IIPermalink

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