Quick reference of the formulas used across the weekly notes, with brief explanations and symbol keys. See weekly notes for full context and examples.
- Sample mean:
$\bar{x} = \frac{1}{n}\sum_{j=1}^n x_j$ . Average of the sample values. - Median (odd
$n$ ):$m = y_{(n+1)/2}$ ; (even$n$ ):$m = \frac{y_{n/2}+y_{n/2+1}}{2}$ . Middle ordered value(s). - Sample variance / SD:
$s^2 = \frac{1}{n-1}\sum_{j=1}^n (x_j-\bar{x})^2$ ,$s=\sqrt{s^2}$ . Uses$n-1$ to estimate population variance from a sample. - Mean absolute deviation:
$\text{m.a.d.} = \frac{1}{n}\sum_{j=1}^n |x_j-\bar{x}|$ . Average absolute distance from the mean. - Empirical CDF:
$F_{\text{emp}}(x)=\frac{1}{n}#{j: x_j\le x}$ . Fraction of data at or below$x$ .- Symbols:
$x_j$ data points,$n$ sample size,$y_j$ ordered values,$#{\cdot}$ count.
- Symbols:
- Kolmogorov axioms:
$P(\varnothing)=0$ ,$P(\Omega)=1$ ,$P(\bigcup A_j)=\sum P(A_j)$ for disjoint$A_j$ . - Combinatorial probability (finite uniform):
$P(E)=\frac{#E}{n}$ . - Inclusion–exclusion (two events):
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$ . - Independence:
$P(A\cap B)=P(A)P(B)$ . - Conditional probability:
$P(A|B)=\frac{P(A\cap B)}{P(B)}$ . - Total probability / Bayes:
$P(A)=\sum_j P(A|B_j)P(B_j)$ ;$P(B_j|A)=\frac{P(A|B_j)P(B_j)}{\sum_k P(A|B_k)P(B_k)}$ .- Symbols:
$A,B,B_j$ events;$\Omega$ sample space;$#E$ count;$n$ outcomes.
- Symbols:
- CDF:
$F_X(x)=P(X\le x)$ (step function for discrete$X$ ). - PMF:
$p_X(x)=P(X=x)$ with$\sum_x p_X(x)=1$ . - PMF/CDF link (discrete): jump at
$x_j$ equals$p_X(x_j)=F_X(x_j)-\lim_{x\to x_j^-}F_X(x)$ .- Symbols:
$X$ random variable;$p_X$ pmf;$F_X$ cdf;$x_j$ support points.
- Symbols:
- Expectation (discrete):
$E\xi=\sum_x x,p_\xi(x)$ ; of a function:$E g(\xi)=\sum_x g(x)p_\xi(x)$ . - Variance:
$\text{Var}(\xi)=E(\xi-E\xi)^2=E\xi^2-(E\xi)^2$ . - Linearity:
$E\left(\sum_j \xi_j\right)=\sum_j E\xi_j$ ;$\text{Var}(c\xi)=c^2\text{Var}(\xi)$ ; for independent$\xi_j$ ,$\text{Var}(\sum_j \xi_j)=\sum_j \text{Var}(\xi_j)$ . - Covariance / correlation:
$\text{Cov}(\xi_1,\xi_2)=E\xi_1\xi_2-E\xi_1E\xi_2$ ;$\rho=\frac{\text{Cov}(\xi_1,\xi_2)}{\sqrt{\text{Var}(\xi_1)\text{Var}(\xi_2)}}$ . - Markov:
$P(\xi\ge x)\le \frac{E\xi}{x}$ for$\xi\ge0$ ; Chebyshev:$P(|\xi-E\xi|\ge x)\le \frac{\text{Var}(\xi)}{x^2}$ .- Symbols:
$\xi,\xi_j$ random variables;$p_\xi$ pmf;$g$ measurable function;$c$ constant;$x>0$ bound.
- Symbols:
- PDF/CDF link:
$F_\xi(x)=\int_{-\infty}^x p_\xi(t),dt$ ; if$F_\xi$ differentiable,$p_\xi(x)=F'_\xi(x)$ . - Uniform
$(a,b)$ :$p_\xi(x)=\frac{1}{b-a}I(a<x<b)$ ;$F_\xi(x)=0$ ($x\le a$ ),$(x-a)/(b-a)$ ($a<x<b$ ),$1$ ($x\ge b$ );$E\xi=\frac{a+b}{2}$ ,$\text{Var}(\xi)=\frac{(b-a)^2}{12}$ . - Normal
$(\mu,\sigma^2)$ pdf:$p_\xi(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$ ; standardization$\eta=\frac{\xi-\mu}{\sigma}\sim N(0,1)$ .- Symbols:
$p_\xi$ pdf;$F_\xi$ cdf;$I(\cdot)$ indicator;$\mu,\sigma$ mean/SD;$a,b$ interval endpoints.
- Symbols:
- Hypergeometric pmf:
$P(\xi=x)=\frac{\binom{b}{x}\binom{N-b}{n-x}}{\binom{N}{n}}$ ;$E\xi=\frac{bn}{N}$ ;$\text{Var}(\xi)=\frac{bn(N-b)(N-n)}{N^2(N-1)}$ . - Exponential pdf:
$p_\xi(x)=\lambda e^{-\lambda x}I(x\ge0)$ ;$E\xi=1/\lambda$ ;$\text{Var}(\xi)=1/\lambda^2$ ; memoryless$P(\xi>t+s|\xi>t)=P(\xi>s)$ . - Cauchy pdf:
$p_\xi(x)=\frac{1}{\pi(1+x^2)}$ (no finite moments). - Expectation via density:
$E\xi=\int x p_\xi(x),dx$ ;$E g(\xi)=\int g(x)p_\xi(x),dx$ . - Density of
$f(\xi)$ (invertible$f$ ):$p_\eta(y)=p_\xi(f^{-1}(y))\left|\frac{d}{dy}f^{-1}(y)\right|I(y\in \text{image})$ .- Symbols:
$N$ population size,$b$ good items,$n$ draws;$\lambda$ rate;$f$ transformation;$\eta=f(\xi)$ .
- Symbols:
- Poisson pmf:
$P(\xi=k)=\frac{\lambda^k e^{-\lambda}}{k!}$ for$k\in\mathbb{N}$ . - Joint pdf to marginal:
$p_{\xi_1}(x_1)=\int p_{\xi}(x_1,x_2),dx_2$ (similarly for other components). - Conditional pdf (continuous):
$p_{\xi_2|\xi_1}(x_2|x_1)=\frac{p_{\xi}(x_1,x_2)}{p_{\xi_1}(x_1)}$ . - Transformation with Jacobian (invertible map
$T$ ):$p_\eta(y)=p_\xi(T^{-1}(y))\left|\det J_{T^{-1}}(y)\right|$ .- Symbols:
$\lambda$ rate;$p_\xi$ joint pdf;$J_{T^{-1}}$ Jacobian of inverse map;$\eta=T(\xi)$ .
- Symbols:
- Multivariate normal pdf:
$p_\xi(x)=\frac{1}{(2\pi)^{d/2}\sqrt{\det K}}\exp\left(-\frac{1}{2}(x-\mu)^T K^{-1}(x-\mu)\right)$ . - Chi-squared (sum of squares of
$d$ iid $N(0,1)$): cdf/pdfs as special case (noted in lecture). - Inversion sampling:
$\xi=F^{-1}(U)$ with$U\sim \text{Uni}(0,1)$ gives cdf$F$ . - Exponential sampling via inversion:
$X=-\frac{1}{\lambda}\ln U$ has$F(x)=1-e^{-\lambda x}$ .- Symbols:
$\mu$ mean vector,$K$ covariance matrix,$d$ dimension;$U$ uniform(0,1).
- Symbols:
- Standardization of sums (CLT-style):
$\frac{S_n-n\mu}{\sigma\sqrt{n}}$ used for normal approximations. - Transport cost intuition (no specific closed form given in notes; see week10 text for details).
- Symbols:
$S_n$ sum,$\mu$ mean,$\sigma$ SD.
- Symbols:
- Bernstein bound (Bernoulli sample mean):
$P\left(\left|\frac{S_n}{n}-p\right|\ge\varepsilon\right)\le 2e^{-n\varepsilon^2/4}$ . - Chebyshev for Bernoulli:
$P\left(\left|\frac{S_n}{n}-p\right|\ge\varepsilon\right)\le \frac{p(1-p)}{n\varepsilon^2}$ . - CLT normalization:
$\frac{S_n-n\mu}{\sigma\sqrt{n}}\xrightarrow{d}N(0,1)$ ; sample mean version$\frac{\bar{\xi}_n-\mu}{\sigma/\sqrt{n}}\xrightarrow{d}N(0,1)$ .- Symbols:
$S_n$ sum of iid with mean$\mu$ , variance$\sigma^2$ ;$p$ Bernoulli parameter;$\varepsilon>0$ tolerance.
- Symbols:
- Shifted exponential-type CI (minimum-based): for pdf
$e^{-(x-\theta)}I(x\ge\theta)$ , with sample minimum$m$ , choose$c_\alpha$ s.t.$P(m-\theta\ge c_\alpha)=\alpha$ ; interval$(m-c_\alpha,,m)$ has level$1-\alpha$ . - Normal mean, variance known: $\bar{\xi}n \pm z{1-\alpha/2}\frac{\sigma}{\sqrt{n}}$ is a
$(1-\alpha)$ CI for$\theta$ . - Normal variance, mean known: based on
$\chi^2$ (see week12 text for exact quantiles).- Symbols: $\bar{\xi}n$ sample mean; $\sigma$ known SD; $z{1-\alpha/2}$ standard normal quantile;
$m$ sample minimum;$\alpha$ significance.
- Symbols: $\bar{\xi}n$ sample mean; $\sigma$ known SD; $z{1-\alpha/2}$ standard normal quantile;
- Risk (squared error):
$R(\hat{\theta},\theta)=E_\theta[(\hat{\theta}-\theta)^2]$ . - Sample median definition: for ordered
$\xi_{(1)}\le\dots\le\xi_{(n)}$ , median is$\xi_{(j+1)}$ if$n=2j+1$ ;$(\xi_{(j)}+\xi_{(j+1)})/2$ if$n=2j$ . - Asymptotic normality (informal):
$\sqrt{n}(\text{median}-\theta)\xrightarrow{d}N\left(0,\frac{1}{4p_\xi(\theta)^2}\right)$ when density at median$\theta$ is positive.- Symbols:
$\hat{\theta}$ estimator;$\theta$ true parameter;$p_\xi$ density at median.
- Symbols:
- Sample moments:
$M_k=\frac{1}{n}\sum_{j=1}^n \xi_j^k$ ; method-of-moments solves$\alpha_k(\theta)=M_k$ . - Likelihood (iid):
$L(\theta)=\prod_{j=1}^n p_\xi(x_j;\theta)$ ; log-likelihood$\ell(\theta)=\sum_{j=1}^n \ln p_\xi(x_j;\theta)$ . - One-sided normal test (variance known): reject
$H_0:\theta=\theta_0$ if $\bar{\xi}n\ge \theta_0+z{1-\alpha}\frac{\sigma}{\sqrt{n}}$.- Symbols:
$\xi_j$ sample;$\alpha_k$ theoretical moments;$\theta$ parameter(s);$p_\xi$ model density/pmf;$\sigma$ known SD;$z_{1-\alpha}$ quantile.
- Symbols:
- Power function (conceptual):
$\beta(\theta)=P_\theta(\text{reject }H_0)$ ; aim for small Type I, large power on$H_1$ . - Unbiased/consistent tests: see week15 notes for specific constructions; formulas follow the same critical-value pattern as week14.
- Symbols:
$\beta(\theta)$ power;$\alpha$ significance;$H_0,H_1$ hypotheses.
- Symbols: