Laws

Law of total expectation (Adam’s Law)

\begin{equation} \mathbb{E}[X] = \mathbb{E}\big[\mathbb{E}[X|N]\big] = \mathbb{E}[\mu N] = \mu\mathbb{E}[N] \end{equation}

e.g.

• $X$ is the total amount of money all customers spend in a shop in a day
• $N$ is the number of customers visited that shop
• $\mu$ is the mean amount of money a customer spends

Law of total variance (Eve’s law)

$$% <![CDATA[ \begin{align} \mathbb{V}[X] &= \mathbb{E}\big[\mathbb{V}[X|N]\big] + \mathbb{V}\big[\mathbb{E}[X|N]\big] \\ &= \mathbb{E}[N\sigma^2] + \mathbb{V}[\mu N] \\ &= \sigma^2\mathbb{E}[N] + \mu^2\mathbb{V}[N] \end{align} %]**>$$

Inequalities

Cauchy-Schwarz inequality

\begin{equation} \mathbb{E}[XY] \le \sqrt{\mathbb{E}[X^2]\mathbb{E}[Y^2]} \end{equation}

Jensen’s inequality

If $f$ is a convex function

\begin{equation} f\big(\mathbb{E}[X]\big) \le \mathbb{E}[f(x)] \end{equation}

Markov’s inequality

\begin{equation} \text{P}\big(\left|X\right| \ge a\big) \le \frac{\mathbb{E}[X]}{a} \end{equation}

Chebyshev’s inequality

\begin{equation} \text{P}\big(\left|X - \mu\right| \gt a \big) \le \frac{\mathbb{V}[X]}{a^2} \end{equation}

where $\mu=\mathbb{E}[X]$, and $a \gt 0$.

Hoeffding’s inequality

TODO, see All of Statistics book

Approximation

Approximate binomial distribution with

• Possion distribution when $n$ is large and $p$ is small ($\to 0$).
• Normal distribution when $n$ is large and $p$ is close to $1/2$.

Population statistics

Suppose there are \(N\) samples in total in the population.

Population mean

\begin{equation} \mu = \frac{1}{N}\sum_{i=1}^{N}X_i \label{eq:populationMean} \end{equation}

Population variance

\begin{equation} \sigma ^2 = \frac{1}{N}\sum_{i=1}^{N}(X_i - \mu) ^2 \label{eq:populationVariance} \end{equation}

$\sigma$ is the population standard deviatin.

Sample statistics

Suppose we take a sample of sample size \(n\) from the population.

Sample mean

Note: \(\overline{X}\) is used instead of \(\mu\).

\begin{equation} \overline{X} = \frac{1}{n}\sum_{i=1}^{n}X_i \label{eq:sampleMean} \end{equation}

Sample variance

\(n-1\) is used instead of \(n\) after Bessel’s correction. The intuition behind such correction is that sample variance tends to underestimate population variance, so we intentionally enlarge it a bit. Please see Bessel’s correction for more details.

Note: \(s^2\) is used instead of \(\sigma ^2\).

\begin{equation} s ^2 = \frac{1}{n - 1}\sum_{i=1}^{n}(X_i - \overline{X}) ^2 \label{eq:sampleVariance} \end{equation}

and $s$ is the sample standard deviation.

Standard error

The full name is the standard error of the mean (SEM), which is the standard deviation of sample mean (\(\overline{X}\)), which is still a random variable (r.v.).

\begin{equation} \textrm{SEM} = \frac{s}{\sqrt n} \label{eq:standardErrorCorrected} \end{equation}

Standardization

Standardization is a common transformation that brings data to be centered at 0 with unit standard deviation.

Let’s denote the transformed value of \(X_i\) as \(X_i'\),

\begin{equation} X’_i = \frac{X_i - \overline{X}}{s} \label{eq:standardization} \end{equation}

Apparently, the mean after standardization \(\overline{x'}\) becomes 0. Let’s calculate the variance of the transformed data,

$$\begin{align} s'^2 & = \frac{\sum_{i=1}^{n}(X'_i) ^2}{n - 1} \nonumber \\ & = \frac{\sum_{i=1}^{n}(\frac{X_i - \overline{X}}{s}) ^2}{n - 1} & \text{replacing}\ X_i'\ \text{with Eq. \eqref{eq:standardization}} \nonumber \\ & = \frac{1}{s^2} \frac{\sum_{i=1}^{n}({X_i - \overline{X}}) ^2}{n - 1} & \text{move constant}\ s^2\ \text{to the front} \nonumber \\ & = \frac{n - 1}{\sum_{i=1}^{n}({X_i - \overline{X}}) ^2} \frac{\sum_{i=1}^{n}({X_i - \overline{X}}) ^2}{n - 1} & \text{replacing }\ s^2\ \text{with Eq. \eqref{eq:sampleVariance}} \nonumber \\ & = 1 \nonumber \\ \label{eq:standarizedSampleStd} \end{align}$$