Statistics

Let $X$ be a random variable with a finite number of outcomes $x_1,x_2,...,x_n$ occurring with probabilities $p_1,p_2,...,p_n$ respectively.

The expectation of $X$ is a generalized notion of weighted average. It is notationally defined as $E[X]=\sum _{i=1}^n{p}_i{x}_i$ .

If all outcomes are equally probable, i.e. $p_1=p_2=...=p_n$ , then the expectation equals the arithmetic mean.

The variance of $X$ is the expectation of the squared deviation of $X$ from its expectation. It is notationally defined as

$${\sigma }^2=E[(X-E[X]{)}^2]$$

The standard deviation of $X$ is defined as $\sigma =\sqrt{{\sigma }^2}$ .

When only a dataset of outcomes of $X$ is available, the underlying metrics are called population metrics, and the inferred metrics are called sample metrics.

The sample mean of an $X$ dataset is defined as

$$\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i.$$

The sample variance of an $X$ dataset is defined as

$$s^2=\frac{1}{n-1}\sum \mathrm{\_}{i=1}^n(x_i-\overline{x}{)}^2$$