Statistics is the function of everything.
Variance & Covariance
- $E(a+by+cz)=a+bE(y)+cE(z)$
- $Var(a+by+cz)=b^2Var(y)+c^2Var(z)+2bccdot Cov(y,z)$
- $Var(sum_{i=1}^{n}a_i y_i) = sum_{i=1}^{n}a_i^2Var(y_i)+2sum_{i<j} a_i a_jCov(y_i,y_j)$
- $Cov(aU+bV,cX+dY)=acCov(U,X)+adCov(U,Y)+bcCov(V,X)+bdCov(V,Y)$
- $Cov(sum_{i=1}^{n}a_ix_i,sum_{i=1}^{n}b_iy_i)=sum_{i=1}^{m}sum_{j=1}^{n}a_ib_jCov(x_i,y_i)$
Random Sample
Define the r.v’s $y_1,y_2…y_n$ are a random sample of size n, if they are all independent and have the same distribution (PMF/PDF), i.e. they are “independent and identically distributed” (iid).
Example 1:
Suppose $y_1, y_2…y_noverset{text{iid}}{sim}Expo(beta)$, the joint density $f(y_1,y_2…y_n)overset{text{indep}}{=}f_{y_1}(y_1)f_{y_2}(y_2)…f_{y_n}(y_n)$
$=frac{1}{beta}e^{-frac{y_1}{beta}}frac{1}{beta}e^{-frac{y_2}{beta}}…frac{1}{beta}e^{-frac{y_n}{beta}}$
$=frac{1}{beta ^n}e^{-frac{sum_{i=1}^{n}y_i}{beta}}, y_i>0 for i=1,2…n$
Define a statistic is a function of the random sample $y_1…y_n$, so a statistic is also a r.v with a distribution, means, variances, etc.
Distribution of statistics are called “sampling distributions.”
Example 2:
Sample mean $bar Y = frac{y_1+y_2+…y_n}{n}$, sample variances $S^2 = frac{1}{n-1}sum_{i=1}^{n}(y_i-bar y)^2$, sample standard deviation (SD) is $S = sqrt{S^2}$.
Properties: let $x_1,x_2…x_n$ be r.v’s with $E(x_i) = mu, Var(x_i)=sigma ^2$
- $E(bar X)=E(frac{x_1+x_2+…x_n}{n})=frac{1}{n}[E(x_1)+…+E(x_n)]=frac{1}{n}cdot nmu = mu$
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