Random variables

\[ {\displaystyle \operatorname {Pr} (X\in S)=P({\omega \in \Omega

X(\omega )\in S})} \]

Definitions of discrete

Probability distribution of discrete r.v., a table.

Expectation of a discrete ${\displaystyle \operatorname {E} [X]=x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}}$

Example, expected value of a roll of a fair six-sided die:

\[\operatorname {E} [X]=1\cdot {\frac {1}{6}}+2\cdot {\frac {1}{6}}+3\cdot {\frac {1}{6}}+4\cdot {\frac {1}{6}}+5\cdot {\frac {1}{6}}+6\cdot {\frac {1}{6}} = \frac{21}{6}= 3.5\]

Variance is mean of squares of deviations:

\[
{\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\operatorname {E} \left[(X-\operatorname {E} [X])^{2}\right]\
&=\operatorname {E} \left[X^{2}-2X\operatorname {E} [X]+\operatorname {E} [X]^{2}\right]\
&=\operatorname {E} \left[X^{2}\right]-2\operatorname {E} [X]\operatorname {E} [X]+\operatorname {E} [X]^{2}\
&=\operatorname {E} \left[X^{2}\right]-\operatorname {E} [X]^{2}\end{aligned}}}
\]

Standard deviation $\sigma = \sqrt{operatorname {E} \left[ X \right]}$ is of the same scale as data. Example: if mean is 3, deviation for 5 is 2.

Properties of mean and variance.

Covariation. For independent $x$ and $y$ their $\operatorname {cov}(x, y) = 0$.
Correlation.
Covariance matrix.

TODO LOW: sample standard variance, https://en.wikipedia.org/wiki/Bessel%27s_correction, http://duramecho.com/Misc/WhyMinusOneInSd.html