Formulas

Probability Density Functions (PDFs)

• Standard Normal PDF\[f(x)= {e^{-x^2} \over \sqrt{2 \pi}}\]
• General Normal PDF\[f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}\]
• Chi-Square PDF\[\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,\]
• Gamma PDF\[x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!\]
• Beta PDF\[ \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!\]
• Student's T PDF\[\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!\]
• Poisson PDF\[\frac{e^{-\lambda} \lambda^k}{k!}\!\]
• Chi PDF\[\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}\]
• Cauchy PDF\[\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\]
• Exponential PDF\[ \lambda e^{-\lambda x},\; x \ge 0\]
• F Distribution PDF\[ \frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) } \]
• Bernoulli PMF\[ f(k;p) \begin{cases} \mbox{p if k = 1,} \\ \mbox{1 - p if k = 0,} \\ \mbox{0 otherwise} \end{cases} \]
• Binomial PMF\[ \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}\]
• Multinomial PMF\[f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}\], where \(x_1+x_2+\cdots+x_k=n\), \(p_1+p_2+\cdots+p_k=1\), and \(0 \le x_i \le n, 0 \le p_i \le 1\).
• Negative Binomial PMF\[ \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k \]
• Negative-Multinomial Binomial PMF\[ P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}} \]
• Geometric PMF\[ \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p \]
• Erlang PDF\[ \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} \]
• Laplace PDF\[ \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) \]
• Continuous Uniform PDF\[ f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases} \]
• Discrete Uniform PMF\[ f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} \]
• Logarithmic PDF\[ f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} \]
• Logistic PDF\[ f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} \]
• Logistic-Exponential PDF\[ f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 \]
• Power Function PDF\[ f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} \]
• Benford's Law\[ P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) \]
• Pareto PDF\[ \frac {kx^k_m} {x^{k+1}} \]
• Non-Central Student T PDF\[ f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx \]
• ArcSine PDF\[ f(x) = \frac{1}{\pi \sqrt{x(1-x)}} \]
• Circle PDF\[ f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] \]
• U-Quadratic PDF\[\alpha \left ( x - \beta \right )^2 \]
• Standard Uniform PDF\[U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases} \]
• Zipf\[\frac{1/(k+q)^s}{H_{N,s}}\]
• Inverse Gamma\[\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)\]
• Fisher-Tippett\[\frac{z\,e^{-z}}{\beta}\!\]
  where \(z = e^{-\frac{x-\mu}{\beta}}\!\)
• Gumbel\[f(x) = e^{-x} e^{-e^{-x}}.\]
• HyperGeometric\[{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}\]
• Log-Normal\[\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]\]
• Gilbrats\[\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]\]
• Hyperbolic Secant\[\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!\]
• Gompertz\[b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]\]
• Standard Cauchy\[ f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!\]
• Rectangular\[ f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!\]
• Beta-Binomial\[ f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!\]
• Negative Hypergeometric\[ f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!\]
• Standard Power\[ f(x; \beta) = \beta x^{\beta - 1} \!\]
• Power_Series\[ f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!\]
• Zeta\[ f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!\]
• Logarithm\[ f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!\]
• Beta_Pascal\[ f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!\]
• Gamma_Poisson\[ f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!\]
• Pascal\[ f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!\]
• Polya\[ f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!\]
• Normal-Gamma\[ f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!\]
• Discrete_Weibull\[ f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!\]
• Log Gamma\[ f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!\]
• Generalized Gamma\[ f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!\]
• Noncentral-Beta\[ f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!\]
• Inverse Gausian\[ f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!\]
• Noncentral_chi-square\[ f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!\]
• Standard Wald\[ f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!\]
• Inverted Beta\[ f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!\]
• Arctangent\[ f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!\]
• Makeham\[ f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta (\kappa^x-1)}{log(\kappa)}). x>0 \!\]
• Hypoexponential\[ f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!\]
• Doubly Noncentral t\[ \!\]
• Hyperexponential\[ f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!\]
• Muth\[ f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!\]
• Error\[ f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!\]
• Minimax\[ f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!\]
• Noncentral F\[ f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!\]
• IDB\[ f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!\]
• Standard Power\[ f(x) = \beta x^{\beta-1}. 0<x<1 \!\]
• Rayleigh\[ f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!\]
• Standard Triangular\[ f(x) = \begin{cases} x+1, -1<x<0 \\ 1 - x, 0 \leq x<1 \end{cases} \!\]
• Doubly noncentral F\[ f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!\]
• Power\[ f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!\]
• Weibull\[ f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!\]
• Log-logistic\[ f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!\]
• TSP\[ f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\ \frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x<b \end{cases} \!\]
• Extreme value\[ f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!\]
• Lomax\[ f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!\]
• von Mises\[ f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!\]
• Generalized Pareto\[ f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!\]
• Triangular\[ f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\ \frac{2(b-x)}{(b-a)(b-m)}, m \le x<b \end{cases}. a<m<b>0 \!\]

Transformations

• Standard Normal to General Normal Transformation\[\mu+\sigma\times X\]
• General Normal to Standard Normal Transformation\[X-\mu \over \sigma\]
• Standard Normal to Chi Transformation\[|\ X |\]
• Standard Normal to Chi-Square Transformation\[\sum_{k=1}^{\nu} X_k^2\]
• Gamma to General Normal Transformation\[\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty\]
• Gamma to Exponential Transformation: The special case of \({\Gamma}(k=1, \theta=1/\lambda)\,\) is equivalent to exponential \(Exp(\lambda)\).
• Gamma to Beta Transformation\[X_1 \over X_1 + X_2\].
• Student's T to Standard Normal Transformation\[n\longrightarrow\infty\]
• Student's T to Cauchy Transformation\[n=1 \ \]
• Cauchy to General Cauchy Transformation\[a + \alpha\times X\]
• General Cauchy to Cauchy Transformation\[a=0; \alpha=1 \ \]
• Fisher's F to Student's T\[\sqrt X \]
• Student's T to Fisher's F\[ X^2 \]
• Bernoulli to Binomial Transformation\[ \sum X_i \] (iid)
• Binomial to Bernoulli Transformation\[\begin{pmatrix} n = 1 \end{pmatrix}\]
• Binomial to General Normal Transformation\[ \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} \]
• Binomial to Poisson Transformation\[ \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} \]
• Multinomial to Binomial Transformation\[ \begin{vmatrix} k=2 \end{vmatrix} \]
• Negative Binomial to Geometric Transformation\[ \begin{pmatrix} r = 1 \end{pmatrix} \]
• Erlang to Exponential Transformation\[ \begin{pmatrix} k = 1 \end{pmatrix} \]
• Erlang to Chi-Square Transformation\[ \begin{pmatrix} \alpha = 2 \end{pmatrix} \]
• Laplace to Exponential Transformation\[\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}\]
• Exponential to Laplace Transformation\[ x_1 - x_2 \ \]
• Beta to Arcsine Transformation\[ \alpha = \beta = \frac{1}{2} \]
• Noncentral Student's T to Normal Transformation\[ Z=\lim_{\nu\to\infty}T \]
• Noncentral Student's T to Student's T Transformation\[ \mu = 0 \ \]
• Standard Uniform to Pareto Transformation\[ \lambda X ^{-1/K} \ \]
• Standard Uniform to Benford Transformation\[ 10^X \ \]
• Standard Uniform to Exponential Transformation\[ n(1-X_{(n)}), n -> \infty \]
• Standard Uniform to Log Logistic Transformation\[ \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} \]
• Standard Uniform to Standard Triangular Transformation\[ X_1 - X_2\]
• Standard Uniform to Logistic Exponential Transformation\[ \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} \]
• Standard Uniform to Beta Transformation: If X has a standard uniform distribution, \( Y = 1 - X^{1/n} \ \) has a beta distribution
• Beta to Standard Uniform Transformation\[ \beta = \gamma = 1 \]
• Continuous Uniform to Standard Uniform Transformation\[ a = 0, b = 1 \ \]
• Pareto to Exponential\[ log(X/\lambda) \ \]
• Logistic Exponential to Exponential\[ \beta = 1 \ \]
• Zipf to Discrete Uniform\[ a = 0, a = 1, b = n \ \]
• Discrete Uniform to Rectangular\[ a = 0, b = n \ \]
• Poisson to Normal\[ \sigma ^2 = \mu , \mu \to \infty \]
• Binomial to Poisson\[ \mu = np, \mu \to \infty \]
• Gamma to Inverted Gamma\[ \frac{1}{X} \]
• Fisher-Tippett to Gumbel\[ \mu = 0, \beta = 1 \ \]
• Hypergeometric to Binomial\[ p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ \]
• Log-Normal to Normal\[ log(X) \ \]
• Normal to Log-Normal\[e^X \ \]
• Log-Normal to Gibrat's\[ \mu = 0, x = 1 \ \]
• Cauchy to Standard Cauchy\[ \gamma = 1, x_0 = 0 \ \]
• Standard Cauchy to Cauchy\[ x_0 + \gamma X \ \]
• Standard Cauchy to Hyperbolic Secant\[ \frac{log|x|}{\pi} \ \]
• Beta to Standard Power\[ \alpha=\beta, \beta=1 \ \]
• Normal to Chi-square\[ (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ \]
• Power series to Pascal\[ A(c)=(1-c)^{-x}, c=1-p \]

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