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 $$f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\ \frac{2(b-x)}{(b-a)(b-m)}, m \le x<b \end{cases}. a<m<b>0 \!$$
• Lévy distribution $$L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2$$
• Modified Power Series distributon $$P(X=x)={a(x)\left\{u(c)\right\}^{x} \over A(c)}$$ where $A(c)=\sum _{x}a(x)\left\{u(c)\right\}^{x} ,a(x)\ge 0$
• Positive binomial distribution $$P(X=x)=\binom{n}{x}{p^{x} q^{n-x} \over (1-q^{n} )}$$ where $x=1,2,...,n$
• Basic Lagrangian distribution of the first kind (BLD1) $$P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } (g(z))^{x} \right]_{z=0}$$ where $g(z)$ is pgf , $g(0)$ is not 0
• General Basic Lagrangian distribution of the first kind (GLD1) $$P(X=0)=f(0) , P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} , x>0$$ Where f(z) and g(z) are pgf , $\left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} >0$ for $x\ge 1$
• Binomial-delta distribution $$P(X=x)={n\over x}\binom[[:Template:Mx]]{x-n}p^{x-n} q^{n+mx-x}$$ for $x\ge n$
• Binomial-Poisson distribution $$P(X=x)=e^{-M} {(Mq^{m} )^{x} \over x!} {}_{2} F{}_{0} [1-x,-mx;{p\over Mq} ]$$ , for $x\ge 0$
• Binomial-negative-binomial distribution $$P(X=x)={\Gamma (k+x)\over x!\Gamma (x)} Q^{-k} \left({Pq^{m} \over Q} \right)^{x} {}_{2} F_{1} [1-x,-mx;1-x-k;{-pQ\over qP} ]$$ for $x\ge 0$
• _Distribution.html Poisson-delta distribution $$P(X=x)={n\over x} {e^{-\theta x} (\theta x)^{x-n} \over (x-n)}$$ for $x\ge n$
• Poisson-Poisson distribution(also called "Generalized Poisson distribution") $$P(X=x)=M(M+\theta x)^{x-1} e^{-(M+\theta x)} /x!$$ for $x\ge 0$
• Poisson-binomial distribution $$P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} npq^{n-1} {}_{2} F_{0} [1-x,1-n;{p\over \theta qx} ] , x\ge 1$$
• Poisson-negative-binomial distribution $$P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} kPQ^{-k-1} {}_{2} F_{0} [1-x,1+k;{-P\over \theta Qx} ] , x\ge 1$$
• Negative-binomial-delta distribution $$P(X=x)={n\over x} {\Gamma (kx+x-1)\over (x-n)!\Gamma (kx)} \left({P\over Q} \right)^{x-n} Q^{-kx} , x\ge n$$
• Negative-binomial-Poisson distribution $$P(X=x)={e^{-M} M^{x} \over x!} Q^{-kx} {}_{2} F_{0} [1-x,kx;-;{-P\over MQ} ]$$ , for $x\ge 0$
• Negative-binomial-binomial distribution $$P(X=0)=q^{n}$$ , $P(X=x)=npq^{n-1} {\Gamma (kx+x-1)\over x!\Gamma (kx)} \left({P\over Q} \right)^{x-1} Q^{-kx} {}_{2} F_{1} [1-x,1-n;2-x-kx;{-pQ\over Pq} ]$ for $x\ge 1$
• Negative-binomial-negative-binomial distribution $$P(X=x)=(Q')^{-M} \left({P'\over Q'Q^{k} } \right)^{x} {\Gamma (M+x)\over x!\Gamma (M)} {}_{2} F_{1} [1-x,kx;1-M-x;{PQ'\over P'Q} ]$$ for $x\ge 1$
• Weight binomial distribution $$P(X=x)=w(x)p_{x} /\sum _{x}^{}w(x)p_{x}$$
• Positive Poisson distribution (conditional Poisson distribution) $$P(X=x)=(e^{\theta } -1)^{-1} \theta ^{x} /x! , x=1,2,......$$
• Left-truncated Poisson distribution $$P(X=x)={e^{-\theta } \theta ^{x} \over x!} \left[1-e^{-\theta } \sum _{j=0}^{r_{1} -1}{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...$$
• Right-truncated Poisson distribution $$P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=0}^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=0,1,...,r_{2}$$
• Doubly-truncated Poisson distribution $$P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=r_{1} }^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...,r_{2} , 0<r_{1} <r_{2}$$
• Misrecorded Poisson distribution $$P(X=0)=\omega +(1-\omega )e^{-\theta }, P(X=x)=(1-\omega ){e^{-\theta } \theta ^{x} \over x!} , x\ge 1$$

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 $$\alpha=\beta, \beta=1 \ $$
• Power series to Pascal $$A(c)=(1-c)^{-x}, c=1-p \ $$
• Gamma Poisson to Pascal $$\alpha=(1-p)/p, \beta=n \ $$
• Poisson to Gamma Poisson $$\mu \sim gamma \ $$
• Discrete uniform to Rectangular $$a=0, b=n\ $$
• beta binomial to rectangular $$a=b=1 \ $$
• beta binomial to negative hypergeometric $$n=n_1, a=n_2, b=n_3 \ $$
• Zipf to Zeta $$n\to\infty\ $$
• Power series to Logarithm $$A(c)=-log(1-c)\ $$
• Power series to Poisson $$A(c)=e^c, \mu=c\ $$
• Pascal to Beta pascal $$p\sim beta\ $$
• pascal to poisson $$\mu=n/p, n\to\infty\ $$
• binomial to beta binomial $$p\sim beta, \mu=np, n\to\infty\ $$
• negative hypergeometric to binomial $$p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ $$
• Polya to Binomial $$\beta=0\ $$
• Pascal to geometric $$n=1 \ $$
• geometric to pascal $$\sum{X_i}\ $$
• discrete weibull to geometric $$\beta=1\ $$
• pascal to normal $$\mu=n(1-p), n\to\infty\ $$
• normal to standard normal $$\mu=0, \sigma=1\ $$
• normal to noncentral_chi-square $$\sum{X_i^2/{\sigma}^2}\ $$
• Normal to Chi-square $$(iid) \sum (\frac{x_i-\mu}{\sigma})^2\ $$
• Beta to Normal $$\beta=\gamma \to \infty \ $$
• Normal to Gamma-normal $$\sigma \sim Inverted \ gamma \ $$
• Standard Normal to Standard Cauchy $$\frac{X_1}{X_2} \ $$
• Inverse Gaussian to Standard normal $$\lambda \to \infty \ $$
• Noncentral chi-square to Chi-square $$\delta=0 \ $$
• Gamma to Log gamma $$log X \ $$
• Generalized gamma to Log normal $$\beta \to \infty \ $$
• Generalized gamma to Gamma $$\gamma=1 \ $$
• Inverse Gaussian to Standard Wald $$\mu=1 \ $$
• Inverse Gaussian to Chi-square $$\lambda(X-\mu)^2/(\mu^2 X)\ $$
• Chi-square to Chi $$\sqrt{X}\ $$
• Chi-square to F $$\frac{X_1/n_1}{X_2/n_2}\ $$
• F to Chi-square $$n_1 X, n_2 \to \infty \ $$
• Exponential to Chi-square $$(iid) \frac{2}{\alpha} \sum {X_i}\ $$
• Chi-square to Exponential $$\alpha=2, n=2 \ $$
• Chi-square to Erlang $$n \ even\ $$
• Gamma to Chi-square $$n=2\beta, \alpha=2 \ $$
• Beta to Standard Uniform $$\beta=\gamma=1 \ $$
• Gamma to Erlang $$\beta=n \ $$
• Gamma to Inverted Beta $$X_1/X_2, \alpha=1 \ $$
• Beta to Inverted Beta $$\frac{X}{1-X} \ $$
• Cauchy to Arctangent $$zero \ truncate \ $$
• Hypoexponential to Erlang $$\vec \alpha=\alpha \ $$
• Exponential to Hypoexponential $$\sum X_i\ $$
• Erlang to Exponential $$n=1 \ $$
• Makeham to Gompertz $$\gamma=0 \ $$
• Doubly noncentral t to Noncentral t $$\gamma=0 \ $$
• Exponential to F $$\alpha=1, X_1/X_2\ $$
• Noncentral F to F $$\delta \to 0\ $$
• Exponential to Hyperexponential $$Mixture\ $$
• Hyperexponential to Exponential $$\vec \alpha=\alpha \ $$
• IDB to Exponential $$\delta=\kappa \to 0, \alpha=1/ \gamma \ $$
• Exponential to Rayleigh $$X^2\ $$
• Weibull to Exponential $$\beta=1 \ $$
• Exponential to Weibull $$X^{1/\beta}\ $$
• Muth to Exponential $$\alpha=1, \kappa \to 0 \ $$
• Standard uniform to Gompertz $$\frac{log[1-(log X)(log \kappa)/\delta]}{log \kappa}\ $$
• Standard uniform to Exponential Power $$[log(1-log(1-X))/\gamma]^{1/\kappa}\ $$
• Error to Laplace $$a=0, b=\alpha/2, c=2\ $$
• Laplace to Error $$\alpha_1=\alpha_2 \ $$
• Standard uniform to log logistic $$\frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ $$
• Standard uniform to Standard triangular $$X_1-X_2 \ $$
• Standard uniform to uniform $$a+(b-a)X \ $$
• Standard uniform to standard power $$X^{1/\beta} \ $$
• Standard power to standard uniform $$\beta=1 \ $$
• Standard uniform to standard power $$X_(n) \ $$
• Minimax to standard power $$\gamma=1 \ $$

• IDB to Rayleigh $$\delta=2/\alpha, \gamma=0 \ $$
• Power to Standard Power $$\alpha=1 \ $$
• Weibull to Rayleigh $$\beta=2 \ $$
• Generalized Pareto to Pareto $$\gamma=0, X+\delta \ $$
• Triangular to standard triangular $$a=-1,b=1,m=0\ $$
• Weibull to Extreme-value $$logX \ $$
• Log logistic to lomax $$\kappa=1 \ $$
• logistic_Distribution.html Lomax to log logistic $$\kappa=1 \ $$
• Log logistic to logistic $$logX \ $$
• TSP to triangular $$n=2 \ $$
• von Mises to Uniform $$\kappa \to 0 \ $$
• Lévy to Cauchy $$\alpha =1 \ $$
• Lévy to Gaussian $$\alpha \to 2$$
• Modified Power Series to Power series $$u(c)=c \ $$
• BLD1 to Geometric $$g(z)=1-p+pz \ $$ where$0<p<1 \ $
• BLD1 to Borel-Tanner $$g(z)=e^{\lambda (z-1)} , 0<\lambda \ $$
• GLD1 to Binomial $$g(z)=1 \ $$ and $f(z)=(q'+p'z)^{n} \ $ where $q'=1-p' \ $ , $0<p'<1 \ $, and n is positive integer.
• GLD1 to Negative binomial $$g(z)=1 \ $$ and $f(z)=(q'+p'z)^{n} \ $ where $q'=1+P \ $ , $0<P \ $ , and $n=-k<0 \ $
• GLD1 to Binomial-delta $$g(z)=(q+pz)^{m} \ $$ , $f(z)=z^{n} \ $ , $mp<1 \ $
• GLD1 to Binomial-Poisson $$: g(z)=(q+pz)^{m} \ $$ , $f(z)=e^{M(z-1)} \ $ , $mp<1 \ $
• GLD1 to Binomial-negative-binomial $$g(z)=(q+pz)^{m} \ $$ , $f(z)=(Q-Pz)^{-k} \ $ , $mp<1 \ $
• GLD1 to Poisson-delta $$g(z)=e^{\theta (z-1)} \ $$ , $f(z)=z^{n} \ $, $\theta <1 \ $
• GLD1 to Poisson-Poisson $$g(z)=e^{\theta (z-1)} , f(z)=e^{M(z-1)} , \theta <1 \ $$
• GLD1 to Poisson-binomial $$g(z)=e^{\theta (z-1)} , f(z)=(q+pz)^{n} , \theta <1 \ $$
• GLD1 to Poisson-negative-binomial $$g(z)=e^{\theta (z-1)} , f(z)=(Q-Pz)^{-k} , \theta <1 \ $$
• GLD1 to Negative-binomial-delta $$g(z)=(Q-Pz)^{-k} , f(z)=z^{n} , kP<1 \ $$
• GLD1 to Negative-binomial-Poisson $$g(z)=(Q-Pz)^{-k} , f(z)=e^{M(z-1)} , kP<1 \ $$
• GLD1 to Negative-binomial-binomial $$g(z)=(Q-Pz)^{-k} , f(z)=(q+pz)^{n} , kP<1 \ $$
• GLD1 to Negative-binomial-negative-binomial $$g(z)=(Q-Pz)^{-k} , f(z)=(Q'-P'z)^{-M} , kP<1 \ $$
• Chi-Square to Poisson $$\left(1-F_{\chi _{2(x+1)}^{2} } (2t/\tau )\right)-\left(1-F_{\chi _{2x}^{2} } (2t/\tau )\right) \ $$ and $\lambda =t/\tau \ $
• Left-truncated Poisson to Positive Poisson $$r_{1} =1 \ $$
• Doubly-truncated Poisson to Right-truncated Poisson $$r_{1} =0 \ $$

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

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