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



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