Formula sheet God help me

Definitions

Acronym	Meaning	Use
PDF	probability density function (continuous)	describes relative likelihood
PMF	probability mass function (discrete)	probability at specific values
CDF	cumulative distribution function	probability ≤ x
CI	confidence interval	estimates parameter range
ANOVA	analysis of variance	tests group mean differences

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Key Symbols

Symbol	Meaning	Use
$n$	sample size	number of observations
$x_i$	i-th data value	individual measurement
$\bar{x}$	sample mean	average of sample
$\mu$	population mean	true average
$s$, $s^2$	sample SD, variance	sample spread
$\sigma$, ${\sigma }^2$	population SD, variance	true spread
$p$	success probability	Bernoulli success rate
$k$	count of successes	number of positive outcomes
$\lambda$	rate parameter	events per unit time
$\alpha$	significance level	probability of Type I error
$z,t$	standard quantiles	critical values
${\chi }^2$	chi-square quantile	variance test value
${\beta }_0,{\beta }_1$	intercept, slope	regression coefficients
$R^2$	coefficient of determination	explained variance fraction

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Basics

Concept	Formula	Use
Mean	$\bar{x}=\frac{1}{n}{\sum }_i{x}_i$	central location
Sample Variance	$s^2=\frac{1}{n-1}\sum (x_i-\bar{x}{)}^2$	measure of dispersion
Population Variance	${\sigma }^2=\frac{1}{n}\sum (x_i-\mu {)}^2$	actual dispersion
PDF	$f(x)$	density shape
CDF (cont.)	$F(x)={\int }_{-\infty }^{x}f(t)dt$	cumulative probability
PMF	$P(X=k)$	discrete probability at k
CDF (disc.)	$F(k)={\sum }_{i=0}^{k}P(X=i)$	cumulative discrete prob.

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Discrete Distributions

Distribution	PMF	CDF	Mean / Var	Use
Binomial $(n,p)$	$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$ or like (P)	$P(X\le x)={\sum }_{i=0}^k\left(\genfrac{}{}{0ex}{}{n}{i}\right)p^i(1-p{)}^{n-i}$	$np$, $np(1-p)$	modeling # successes in n trials
Poisson $(\lambda )$	$P(X=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!}$	$F(k)={\sum }_{i=0}^k\frac{{\lambda }^i{e}^{-\lambda }}{i!}$	$\lambda$, $\lambda$	counts of rare events
Binomial	$\left[\begin{array}{c}n\\ x\end{array}\right]=\frac{n!}{(n-x)!x!}$

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Continuous Distributions

Distribution	PDF	CDF	$E$, $Var$	Use
Normal $(\mu ,{\sigma }^2)$	$\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$	$\Phi (\frac{x-\mu }{\sigma })$	$\mu$, ${\sigma }^2$	symmetric spread around mean
Exponential $(\lambda )$	$\lambda e^{-\lambda x},x\ge 0$	$1-e^{-\lambda x}$	$1/\lambda$, $1/{\lambda }^2$	time between Poisson events
Lognormal $(\mu ,{\sigma }^2)$	$\frac{1}{x\sigma \sqrt{2\pi }}{e}^{-\frac{(\ln x-\mu {)}^2}{2{\sigma }^2}}$	$\Phi (\frac{\ln x-\mu }{\sigma })$	$e^{\mu +\frac{1}{2}{\sigma }^2}$, $(e^{{\sigma }^2}-1)e^{2\mu +{\sigma }^2}$	modeling skewed positive data
Gamma $(\alpha ,\beta )$	$\frac{x^{\alpha -1}{e}^{-x/\beta }}{\Gamma (\alpha ){\beta }^{\alpha }}$	$\frac{1}{\Gamma (\alpha )}\gamma (\alpha ,x/\beta )$	$\alpha \beta$, $\alpha {\beta }^2$	sum of $\alpha$ exponentials
Weibull $(k,\lambda )$	$\frac{k}{\lambda }(\frac{x}{\lambda }{)}^{k-1}{e}^{-(x/\lambda {)}^k}$	$1-e^{-(x/\lambda {)}^k}$	$\lambda \Gamma (1+\frac{1}{k})$, ${\lambda }^2[\Gamma (1+\frac{2}{k})-\Gamma (1+\frac{1}{k}{)}^2]$	life/failure time distribution

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Parameter Estimation

Estimate	Formula	Use
CI for mean	$\bar{x}\pm t_{n-1,\alpha /2}\frac{s}{\sqrt{n}}$	interval for true mean
CI for variance	$(\frac{(n-1)s^2}{{\chi }_{1-\alpha /2}^2},\frac{(n-1)s^2}{{\chi }_{\alpha /2}^2})$	interval for true variance
CI for proportion	$\hat{p}\pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$	interval for true proportion

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Hypothesis Testing

Test	Statistic	Decision Rule	Use
Z-test	$Z=\frac{\bar{x}-{\mu }_0}{\sigma /\sqrt{n}}$	reject if $|Z|>z_{\alpha /2}$	test mean with known $\sigma$
T-test	$T=\frac{\bar{x}-{\mu }_0}{s/\sqrt{n}}$	reject if $|T|>t_{n-1,\alpha /2}$	test mean with unknown $\sigma$
p-value	$p=P(|Z|>|z|)$	reject if $p<\alpha$	quantify evidence against $H_0$

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Error Propagation

Rule	Formula	Use
Addition / Subtraction	$Z=X\pm Y,\Delta Z=\Delta X+\Delta Y$	combine absolute errors
Multiplication / Division	$Z=X\cdot Y\text{ or }Z=X/Y,\frac{\Delta Z}{\Vert Z\Vert }=\frac{\Delta X}{\Vert X\Vert }+\frac{\Delta Y}{\Vert Y\Vert }$	Z
Power	$Z=X^n,\frac{\Delta Z}{\Vert Z\Vert }=|n|\frac{\Delta X}{\Vert X\Vert }$	Z
General Function	$Z=f(X_1,\dots ,X_k),\Delta Z=\sqrt{{\sum }_{i=1}^k(\frac{\partial f}{\partial X_i}\Delta X_i{)}^2}$	derivative propagation
Outlier Criterion (Chauvenet)	$|x-\bar{x}|>k\sigma$	detect outliers

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Regression Analysis

Concept	Formula	Use
Slope	${\hat{\beta }}_1=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x}{)}^2}$	effect per unit change
Intercept	${\hat{\beta }}_0=\bar{y}-{\hat{\beta }}_1\bar{x}$	value when x=0
$R^2$	$1-\frac{\sum e_i^2}{\sum (y_i-\bar{y}{)}^2}$	proportion variance explained
ANOVA F	$F=\frac{SSR/1}{SSE/(n-2)}$	test overall regression fit