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 $$f(x; n,\delta) = \sum_{k=0}^{\infty}\ frac{exp(-\delta/2) (\delta/2)^k}{k!} \frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!$$
• Standard Wald $$f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0)\!$$
• Inverted Beta $$f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!$$

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 $$\alpha=\beta, \beta=1 \ $$

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• SOCR Home page: http://www.socr.ucla.edu

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