This is a summary of common probability distributions in engineering and statistics. This chart has the plots of the pdf or pmf (LaTeX source):

# discrete distributions

binomial distribution

- A big urn with balls in either white or black color. Drawing a white ball from urn has probability \(x\) (i.e., black ball has probability \(1-x\)). If we draw \(n\) balls from urn with replacement, the probability of getting \(k\) white balls:

Poisson distribution

- Balls are added to the urn at rate of \(\lambda\) per unit time, under exponential distribution. The probability of having \(k\) balls added to the urn within time \(t\):

geometric distribution

- The probability of have to draw \(k\) balls to see the first white ball being drawn:

negative binomial distribution

- same as the distribution of the sum of \(r\) iid geometric random variable
- negative binomial approximates Poisson with \(\lambda = r(1-x)\) with large \(r\) and \(x\approx 1\)
- Drawing balls from the urn. If we have to draw \(k\) balls to see the \(r\)-th white ball (we have drawn \(r\) white balls and \(k-r\) black balls). The probability of \(k\):

hypergeometric distribution

- A urn with \(N\) balls (finite) and \(K\) balls amongst are white. Draw, without replacement, \(n\) balls from the urn to get \(k\) white balls:

# continuous distributions

uniform distribution

- extreme of flattened distribution
- with upper and lower bounds

triangular distribution

- with upper and lower bounds

normal distribution

- strong tendency for data at central value; symmetric, equally likely for positive and negative deviations from its central value
- frequency of deviations falls off rapidly as we move further away from central value

- \[X_1 \sim N(\mu_1, \sigma^2_1); X_2 \sim N(\mu_2, \sigma^2_2) \to X_1+X_2 \sim N(\mu_1+\mu_2, \sigma_1^2+\sigma_2^2)\]
- approximation to Poisson distribution: if \(\lambda\) is large, Poisson distribution approximates normal with \(\mu=\sigma^2=\lambda\)
- approximation to binomial distribution: if \(n\) is large and \(x\approx \frac{1}{2}\), binomial distribution approximates normal with \(\mu=nx\) and \(\sigma^2=nx(1-x)\)
- approximation to beta distribution: if \(\alpha\) and \(\beta\) are large, beta distribution approximates normal with \(\mu=\frac{\alpha}{\alpha+\beta}\) and \(\sigma^2=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)

Laplace distribution

- absolute difference from mean compared to squared difference in normal distribution
- longer (fatter) tails, higher kurtosis (flattened peak)
- pdf:

logistic distribution

- symmetric, with longer tails and higher kurtosis than normal distribution
- logistic distribution has finite mean \(\mu\) and variance defined
- \[X\sim U(0,1) \to \mu+s[\log(X)-\log(1-X)] \sim \textrm{Logistic}(\mu,s)\]
- \[X\sim \textrm{Exp}(1) \to \mu+s\log(e^X-1) \sim \textrm{Logistic}(\mu,s)\]
- logistic pdf:

Cauchy distribution

- symmetric, with longer tails and higher kurtosis than normal distribution
- Cauchy distribution has mean and variance undefined, but mean & mode at \(\mu\)
- \[X,Y\sim N(\mu,\sigma^2) \to X/Y \sim \textrm{Cauchy}(\mu,\sigma^2)\]
- Cauchy pdf:

lognormal distribution

- \(\log(X)\sim N(\mu,\sigma^2)\), positively skewed
- parameterised by shape (\(\sigma\)), scale (\(\mu\), or median), shift (\(\theta\))
- \(\mu=0, \theta=1\) is standard lognormal distribution
- as \(\sigma\) rises, the peak shifts to left and skewness increases

- sum of two lognormal random variable is a lognormal random variable with \(\mu=\mu_1+\mu_2\) and \(\sigma^2=\sigma_1^2+\sigma_2^2\)

Pareto distribution

- power law probability distribution
- continuous counterpart of Zipf’s law
- positively skewed, no negative tail, peak at \(x=0\)

gamma distribution

- support for \(x\in(0,\infty)\), positive skewness (lean left)
- decreasing \(\alpha\) will push distribution towards the left; at low \(\alpha\), left tail will disappear and distribution will resemble exponential
- models the time to the \(\alpha\)-th Poisson arrival with arrival rate \(\beta\)
- gamma pdf (\(\alpha=1\) becomes exponential pdf with rate \(\beta\)):

Weibull distribution

- support for \(x\in(0,\infty)\), positive skewness (lean left)
- decreasing \(k\) will push distribution towards the left; at low \(k\), left tail will disappear and distribution will resemble exponential
- If \(W\sim\textrm{Weibull}(k,\lambda)\), then \(X=W^k \sim \textrm{Exp}(1/\lambda^k)\)
- Weibull pdf (\(k=1\) becomes exponential pdf with rate \(1/\lambda\)):

Erlang distribution

- \[X_i\sim\textrm{Exp}(\lambda) \to \sum_{i=1}^k X_i \sim \textrm{Erlang}(k, \lambda)\]
- arise from teletraffic engineering: time to \(k\)-th call

beta distribution

- support for \(x\in(0,1)\)
- allows negative skewness
- two shape parameters \(p\) and \(q\), and lower- and upper-bounds on data (\(a\) and \(b\))

extreme value distribution (i.e. Gumbel minimum distribution)

- negatively skewed
- Gumbel maximum distribution, \(f(-x;-\mu,\beta)\), is positively skewed
- Limiting distribution of the max/min value of \(n\to\infty\) iid samples from \(\textrm{Exp}(\lambda)\) with \(\lambda = 1/\beta\)
- standard cdf: \(F(x)=1-\exp(-e^x)\)

Rayleigh distribution

- positively skewed
- modelling the \(L^2\)-norm of two iid normal distribution with zero mean (e.g., orthogonal components of a 2D vector)

Maxwell-Boltzmann distribution

- positively skewed
- 3D counterpart of Rayleigh distribution
- arise from thermodynamic: probability of a particle in speed \(v\) if temperature is \(T\)

Chi-squared distribution

- distribution of the sum of the square of \(k\ge 1\) i.i.d. standard normal random variables
- mean \(k\), variance \(2k\)
- PDF with \(k\) degrees of freedom:

F-distribution

- Distribution of a random variable defined as the ratio of two independent \(\chi^2\)-distributed random variables, with degrees of freedom \(d_1\) and \(d_2\) respectively
- Commonly used in ANOVA
- PDF, with degrees of freedom \(d_1\) and \(d_2\), involves beta function \(B(\alpha,\beta)\):

Student’s t distribution

- Distribution of
*normalized*sample mean of \(n=k+1\) observations from a normal distribution, \(\frac{\bar{X}-\mu}{S/\sqrt{n}}\) - Equivalently, this is the distribution of \(\frac{x}{\sqrt{y/r}}\) for \(x\) is standard normal and \(y\) is chi-square with \(r\) degrees of freedom
- t distribution with \(n=1\) is Cauchy distribution
- PDF with degree of freedom \(k\):

# test of fit for distributions

Kolmogorov-Smirnov test (K-S test, on cumulative distribution function \(F(x)\))

\[D_n = \sup_x | F_n(x) - F(x) |\]- if sample comes from distribution, \(D_n\) converges to 0 a.s. as number of samples \(n\) goes to infinity

Shapiro-Wilk test

\[W = \frac{\sum_{i=1}^n a_i x_i}{\sum_{i=1}^n (x_i - \bar{x})^2}\]- test of normality in frequentist statistics (i.e. for \(x_i\) in normal distribution)
- \(\bar{x} = \frac{1}{n}(x_1 + \cdots + x_n)\) is the sample mean
- \((a_1,\cdots,a_n) = m^T V^{-1} (m^T V^{-1}V^{-1} m)^{-1/2}\) where \(m\) is vector of expected values of the order statistics from normal distribution and \(V\) the covariance matrix of those order statistics

Anderson-Darling test

\[A^2 = n \int_{-\infty}^{\infty} \frac{(F_n(x)-F(x))^2}{F(x)(1-F(x))} dF(x)\]- test whether a sample comes from a specified distribution
- \(A^2\) is weighted distance between \(F_n(x)\) and \(F(x)\), with more weight on tails of the distribution

Pearson’s \(\chi^2\) test

\[\chi^2 = \sum_{i=1}^n \frac{(O_i - E_i)^2}{E_i}\]- test for categories fit a distribution: checking observed frequency \(O_i\) against expected frequency \(E_i\) according to distribution for each of \(n\) categories
- degree of freedom: \(n\) minus number of parameters of the fitted distribution

# Reference

Lawrence M. Leemis and Jacquelyn T. McQuestion. Univariate Distribution Relationships, Am Stat, 62(1) pp.45–53, 2008, DOI: 10.1198/000313008X270448

Aswath Damodaran. Probabilistic approaches: Scenario analysis, decision trees and simulations (PDF, the appendix is also available separately) and includes the following chart for choosing a distribution: