Random Variable

A random variable is a numerical variable whose outcome is the result of a random process (i.e. we don't know what will happen for certain). Notably, the numerical interpretation of the outcome of an experiment is a random variable. They come in two types, deserving their own pages. See

Statistics

Random variables has several statistics that we care about. See pages Continuous Random Variable and Discrete Random Variable for how to calculate these values.

When we are interested in the average outcome of a random variable, we look at the expected value (mean): a weighted average of the possible outcomes.

When we are interested in the variability of a random variable, we look at the variance and the standard deviation: the latter being the expected difference from mean, and the former being the square of the latter (not really used for interpretation, more for math).

Properties of Statistics

I'm just gonna start writing equations.

$E(c)=c$

$E(aX)=aE(X)$

$E(aX+c)=aE(X)+c$

$Var(X)=E((X-\mu )^{2})=E(X^{2})-E(X)^{2}$

Cool name for this: the law of the unconscious statistician for how obvious it is $E(g(X))=\sum g(x_{i})P(X=x_{i})$

$Var(c)=0$

$Var(aX)=a^{2}Var(X)$

$Var(aX+c)=a^{2}Var(X)$

Linear Combinations

A linear combination of random variables $X$ and $Y$ is given as

$aX+bY$

where a and b are constants. They are not bivariate

The expectation is

$E(aX+bY)=aE(X)+bE(Y)$

whereas the variance is

$Var(aX+bY)=a^{2}Var(X)+abCov(X,Y)+b^{2}Var(Y)$

I'm sleepy so I'm not gonna derive this one.