Formulas
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==Probability Density Functions (PDFs)== | ==Probability Density Functions (PDFs)== | ||
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\lambda e^{-\lambda x},\; x \ge 0</math> | \lambda e^{-\lambda x},\; x \ge 0</math> | ||
* [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \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) } </math> | * [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html F Distribution] PDF: <math> \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) } </math> | ||
- | * [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF: <math> | + | * [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli] PMF:<math> f(k;p) \begin{cases} \mbox{p if k = 1,} \\ |
+ | \mbox{1 - p if k = 0,} \\ | ||
+ | \mbox{0 otherwise} \end{cases} </math> | ||
* [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math> | * [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: <math> \begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: <math>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}</math>, where <math>x_1+x_2+\cdots+x_k=n</math>, <math>p_1+p_2+\cdots+p_k=1</math>, and <math>0 \le x_i \le n, 0 \le p_i \le 1</math>. | ||
* [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math> | * [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: <math> \begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/NegativeMultiNomial_Distribution.html Negative-Multinomial Binomial] PMF: <math> 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!}} </math> | ||
* [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math> | * [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: <math> \begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p </math> | ||
* [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math> | * [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: <math> \frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!} </math> | ||
* [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math> | * [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: <math> \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: <math> 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} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: <math> f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: <math> f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0 </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: <math> f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: <math> P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d}) </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: <math> \frac {kx^k_m} {x^{k+1}} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: <math> 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 </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ArcSine_Distribution.html ArcSine] PDF: <math> f(x) = \frac{1}{\pi \sqrt{x(1-x)}} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Circle_Distribution.html Circle] PDF: <math> f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r] </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/UQuadratic_Distribution.html U-Quadratic] PDF: <math>\alpha \left ( x - \beta \right )^2 </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform] PDF: <math>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} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/InverseGamma_Distribution.html Inverse Gamma]: <math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math><br /> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Gumbel_Distribution.html Gumbel]: <math>f(x) = e^{-x} e^{-e^{-x}}.</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html HyperGeometric]: <math>{{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal]: <math>\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Gilbrats_Distribution.html Gilbrats]: <math>\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Hyperbolic Secant]:<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Gompertz]: <math>b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy]: <math> f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Rectangular.html Rectangular]: <math> f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!</math> | ||
+ | * [http://mathworld.wolfram.com/BetaBinomialDistribution.html Beta-Binomial]: <math> 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)\!</math> | ||
+ | * [http://planetmath.org/encyclopedia/NegativeHypergeometricDistribution.html Negative Hypergeometric]: <math> 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)\!</math> | ||
+ | * [[AP_Statistics_Curriculum_2007_Power_Standard |Standard Power]]: <math> f(x; \beta) = \beta x^{\beta - 1} \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Power_Series.html Power_Series]: <math> f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Zeta.html Zeta]: <math> f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Logarithm.html Logarithm]: <math> f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Beta_Pascal(factorial).html Beta_Pascal]: <math> 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) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Gamma_Poisson.html Gamma_Poisson]: <math> f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Pascal.html Pascal]: <math> f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Polya.html Polya]: <math> 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\}) \!</math> | ||
+ | * [http://en.wikipedia.org/wiki/Normal-gamma_distribution Normal-Gamma]: <math> 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) \!</math> | ||
+ | * [http://en.wikipedia.org/wiki/Discrete_Weibull_distribution Discrete_Weibull]: <math> f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/LogGamma.html Log Gamma]: <math> f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/GeneralizedGamma.html Generalized Gamma]: <math> f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Noncentral-Beta.html Noncentral-Beta]: <math> 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). \!</math> | ||
+ | * [http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution Inverse Gausian]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Noncentral_chi-square.html Noncentral_chi-square]: <math> 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})}. \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/StandardWald.html Standard Wald]: <math> f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/InvertedBeta.html Inverted Beta]: <math> f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Arctangent.html Arctangent]: <math> f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Makeham.html Makeham]: <math> f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta | ||
+ | (\kappa^x-1)}{log(\kappa)}). x>0 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Hypoexponential.html Hypoexponential]: <math> 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 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-t_Distribution.html Doubly Noncentral t]: <math> \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Hyperexponential_Distribution.html Hyperexponential]: <math> f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Muth.html Muth]: <math> f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Error.html Error]: <math> f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Minimax.html Minimax]: <math> f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Noncentral-F.html Noncentral F]: <math> 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 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/IDB.html IDB]: <math> f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Standard-power.html Standard Power]: <math> f(x) = \beta x^{\beta-1}. 0<x<1 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Rayleigh.html Rayleigh]: <math> f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Standard-triangular.html Standard Triangular]: <math> f(x) = \begin{cases} x+1, -1<x<0 \\ | ||
+ | 1 - x, 0 \leq x<1 \end{cases} \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Doubly-noncentral-F.html Doubly noncentral F]: <math> 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 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Power_Distribution.html Power]: <math> f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Weibull]: <math> f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Log-logistic]: <math> f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/TwoSidedPower_Distribution.html TSP]: <math> 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} \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Extreme value]: <math> f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Lomax]: <math> f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/VonMises_Distribution.html von Mises]: <math> f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Generalized-Pareto_Distribution.html Generalized Pareto]: <math> f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Triangle_Distribution.html Triangular]: <math> 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 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Kolmogorov_Distribution.html Kolmogorov-Smirnov]: <math> \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Exponential-power_Distribution.html Exponential Power]: <math> f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: <math> 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 </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Modified-Power-Series_Distribution.html Modified Power Series distributon]: <math> P(X=x)={a(x)\left\{u(c)\right\}^{x} \over A(c)} </math> where <math> A(c)=\sum _{x}a(x)\left\{u(c)\right\}^{x} ,a(x)\ge 0 </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Positive-binomial_Distribution.html Positive binomial distribution]: <math> P(X=x)=\binom{n}{x}{p^{x} q^{n-x} \over (1-q^{n} )} </math> where <math> x=1,2,...,n </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Basic-Lagrangian-distribution-of-the-first-kind.html Basic Lagrangian distribution of the first kind (BLD1)]: <math> P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } (g(z))^{x} \right]_{z=0} </math> where <math> g(z) </math> is pgf , <math> g(0) </math> is not 0 | ||
+ | * [http://socr.ucla.edu/htmls/dist/General-Basic-Lagrangian-distribution-of-the-first-kind.html General Basic Lagrangian distribution of the first kind (GLD1)]: <math> 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</math> Where f(z) and g(z) are pgf , <math>\left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} >0</math> for <math>x\ge 1</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Binomial-delta_Distribution.html Binomial-delta distribution]: <math> P(X=x)={n\over x}\binom{{mx}}{x-n}p^{x-n} q^{n+mx-x} </math> for <math>x\ge n</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Binomial-Poisson_Distribution.html Binomial-Poisson distribution]: <math> P(X=x)=e^{-M} {(Mq^{m} )^{x} \over x!} {}_{2} F{}_{0} [1-x,-mx;{p\over Mq} ] </math> , for <math>x\ge 0 </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Binomial-negative-binomial_Distribution.html Binomial-negative-binomial distribution]: <math> 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} ] </math> for <math>x\ge 0</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Poisson-delta _Distribution.html Poisson-delta distribution]: <math> P(X=x)={n\over x} {e^{-\theta x} (\theta x)^{x-n} \over (x-n)} </math> for <math>x\ge n </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Poisson-Poisson_Distribution.html Poisson-Poisson distribution(also called "Generalized Poisson distribution")]: <math> P(X=x)=M(M+\theta x)^{x-1} e^{-(M+\theta x)} /x! </math> for <math>x\ge 0 </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Poisson-binomial_Distribution.html Poisson-binomial distribution]: <math> 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</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Poisson-negative-binomial_Distribution.html Poisson-negative-binomial distribution]: <math> 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</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Negative-binomial-delta_Distribution.html Negative-binomial-delta distribution]: <math> 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 </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Negative-binomial-Poisson_Distribution.html Negative-binomial-Poisson distribution]: <math> P(X=x)={e^{-M} M^{x} \over x!} Q^{-kx} {}_{2} F_{0} [1-x,kx;-;{-P\over MQ} ] </math> , for <math>x\ge 0</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Negative-binomial-binomial_Distribution.html Negative-binomial-binomial distribution]: <math> P(X=0)=q^{n} </math> , <math>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} ] </math> for <math>x\ge 1</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Negative-binomial-negative-binomial_Distribution.html Negative-binomial-negative-binomial distribution]: <math> 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} ] </math> for <math>x\ge 1</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Weight-binomial_Distribution.html Weight binomial distribution]: <math> P(X=x)=w(x)p_{x} /\sum _{x}^{}w(x)p_{x}</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Positive-Poisson_Distribution.html Positive Poisson distribution (conditional Poisson distribution)]: <math> P(X=x)=(e^{\theta } -1)^{-1} \theta ^{x} /x! , x=1,2,......</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Left-truncated-Poisson_Distribution.html Left-truncated Poisson distribution]: <math> 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,...</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Right-truncated-Poisson_Distribution.html Right-truncated Poisson distribution]: <math> P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=0}^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=0,1,...,r_{2}</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Doubly-truncated-Poisson_Distribution.html Doubly-truncated Poisson distribution]: <math> 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}</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Misrecorded-Poisson_Distribution.html Misrecorded Poisson distribution]: <math> P(X=0)=\omega +(1-\omega )e^{-\theta }, P(X=x)=(1-\omega ){e^{-\theta } \theta ^{x} \over x!} , x\ge 1</math> | ||
==Transformations== | ==Transformations== | ||
Line 37: | Line 134: | ||
* [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math> | * [http://en.wikipedia.org/wiki/Student%27s_t_distribution Student's T to Fisher's F]: <math> X^2 </math> | ||
* [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid) | * [http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli to Binomial Transformation]: <math> \sum X_i </math> (iid) | ||
- | * [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math> n = 1 </math> | + | * [http://en.wikipedia.org/wiki/Bernoulli_distribution Binomial to Bernoulli Transformation]: <math>\begin{pmatrix} n = 1 \end{pmatrix}</math> |
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math> | * [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to General Normal Transformation]: <math> \begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix} </math> | ||
* [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math> | * [http://en.wikipedia.org/wiki/Binomial_distribution Binomial to Poisson Transformation]: <math> \begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix} </math> | ||
+ | * [[AP_Statistics_Curriculum_2007_Distrib_Multinomial | Multinomial to Binomial Transformation]]: <math> \begin{vmatrix} k=2 \end{vmatrix} </math> | ||
+ | * [http://en.wikipedia.org/wiki/NegativeBinomial_distribution Negative Binomial to Geometric Transformation]: <math> \begin{pmatrix} r = 1 \end{pmatrix} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Exponential Transformation]: <math> \begin{pmatrix} k = 1 \end{pmatrix} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang to Chi-Square Transformation]: <math> \begin{pmatrix} \alpha = 2 \end{pmatrix} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace to Exponential Transformation]: <math>\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Exponential to Laplace Transformation]: <math> x_1 - x_2 \ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Beta_distribution Beta to Arcsine Transformation]: <math> \alpha = \beta = \frac{1}{2} </math> | ||
+ | * [http://en.wikipedia.org/wiki/Student's_t_distribution Noncentral Student's T to Normal Transformation]: <math> Z=\lim_{\nu\to\infty}T </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Noncentral Student's T to Student's T Transformation]: <math> \mu = 0 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Pareto Transformation]: <math> \lambda X ^{-1/K} \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Benford Transformation]: <math> 10^X \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Exponential Transformation]: <math> n(1-X_{(n)}), n -> \infty </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Log Logistic Transformation]: <math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/k} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Standard Triangular Transformation]: <math> X_1 - X_2</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Logistic Exponential Transformation]: <math> \frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard Uniform to Beta Transformation]: If X has a standard uniform distribution, <math> Y = 1 - X^{1/n} \ </math> has a beta distribution | ||
+ | * [http://en.wikipedia.org/wiki/Beta_distribution Beta to Standard Uniform Transformation]: <math> \beta = \gamma = 1 </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform to Standard Uniform Transformation]: <math> a = 0, b = 1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto to Exponential]: <math> log(X/\lambda) \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic Exponential to Exponential]: <math> \beta = 1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ZipfMandelbrot_Distribution.html Zipf to Discrete Uniform]: <math> a = 0, a = 1, b = n \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform to Rectangular]: <math> a = 0, b = n \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson to Normal]: <math> \sigma ^2 = \mu , \mu \to \infty </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial to Poisson]: <math> \mu = np, \mu \to \infty </math> | ||
+ | * [http://en.wikipedia.org/wiki/Gamma_distribution Gamma to Inverted Gamma]: <math> \frac{1}{X} </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/FisherTippett_Distribution.html Fisher-Tippett to Gumbel]: <math> \mu = 0, \beta = 1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/HyperGeometric_Distribution.html Hypergeometric to Binomial]: <math> p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Normal]: <math> log(X) \ </math> | ||
+ | * [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Prob Normal to Log-Normal]: <math>e^X \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Log-Normal to Gibrat's]: <math> \mu = 0, x = 1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Standard Cauchy]: <math> \gamma = 1, x_0 = 0 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Cauchy to Cauchy]: <math> x_0 + \gamma X \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/HyperbolicSecant_Distribution.html Standard Cauchy to Hyperbolic Secant]: <math> \frac{log|x|}{\pi} \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Standard_Power.html Beta to Standard Power]: <math> \alpha=\beta, \beta=1 \ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Pascal Power series to Pascal]: <math> A(c)=(1-c)^{-x}, c=1-p \ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Pacal Gamma Poisson to Pascal]: <math> \alpha=(1-p)/p, \beta=n \ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Gamma_Poisson Poisson to Gamma Poisson]: <math> \mu \sim gamma \ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Rectangular Discrete uniform to Rectangular]: <math> a=0, b=n\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Rectangualr beta binomial to rectangular]: <math> a=b=1 \ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Negative_hypergeometric beta binomial to negative hypergeometric]: <math> n=n_1, a=n_2, b=n_3 \ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Zeta Zipf to Zeta]: <math> n\to\infty\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Logarithm Power series to Logarithm]: <math> A(c)=-log(1-c)\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Poisson Power series to Poisson]: <math> A(c)=e^c, \mu=c\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Beta_Pascal Pascal to Beta pascal]: <math> p\sim beta\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Poisson pascal to poisson]: <math> \mu=n/p, n\to\infty\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Beta_binomial binomial to beta binomial]: <math> p\sim beta, \mu=np, n\to\infty\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Binomial negative hypergeometric to binomial]: <math> p=n_1/n_3, n_3\to\infty, n_1\to\infty,n_2=n\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Binomial Polya to Binomial]: <math> \beta=0\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Geometric Pascal to geometric]: <math> n=1 \ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Pascal geometric to pascal]: <math> \sum{X_i}\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Geometric discrete weibull to geometric]: <math> \beta=1\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Normal pascal to normal]: <math> \mu=n(1-p), n\to\infty\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/standard_normal normal to standard normal]: <math> \mu=0, \sigma=1\ </math> | ||
+ | * [http://en.wikipedia.org/wiki/Noncentral_chi-square normal to noncentral_chi-square]: <math> \sum{X_i^2/{\sigma}^2}\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Normal to Chi-square]: <math> (iid) \sum (\frac{x_i-\mu}{\sigma})^2\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Beta to Normal]: <math> \beta=\gamma \to \infty \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Normal to Gamma-normal]: <math> \sigma \sim Inverted \ gamma \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Standard Normal to Standard Cauchy]: <math> \frac{X_1}{X_2} \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html Inverse Gaussian to Standard normal]: <math> \lambda \to \infty \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Noncentral chi-square to Chi-square]: <math> \delta=0 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Log gamma]:<math> log X \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/LogNormal_Distribution.html Generalized gamma to Log normal]:<math> \beta \to | ||
+ | \infty \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Generalized gamma to Gamma]:<math> \gamma=1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/InverseGaussian_Distribution.html Inverse Gaussian to Standard Wald]:<math> \mu=1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Inverse Gaussian to Chi-square]:<math> \lambda(X-\mu)^2/(\mu^2 X)\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Chi_Distribution.html Chi-square to Chi]:<math> \sqrt{X}\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Chi-square to F]:<math> \frac{X_1/n_1}{X_2/n_2}\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html F to Chi-square]:<math> n_1 X, n_2 \to \infty \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Exponential to Chi-square]:<math> (iid) \frac{2}{\alpha} \sum {X_i}\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Chi-square to Exponential]:<math> \alpha=2, n=2 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Chi-square to Erlang]:<math> n \ even\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Gamma to Chi-square]:<math> n=2\beta, \alpha=2 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ChiSquare_Distribution.html Beta to Standard Uniform]:<math> \beta=\gamma=1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Gamma to Erlang]:<math> \beta=n \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma to Inverted Beta]:<math> X_1/X_2, \alpha=1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Beta_Distribution.html Beta to Inverted Beta]:<math> \frac{X}{1-X} \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Cauchy to Arctangent]:<math> zero \ truncate \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Hypoexponential to Erlang]:<math> \vec \alpha=\alpha \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Exponential to Hypoexponential]:<math> \sum X_i\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Erlang to Exponential]:<math> n=1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Makeham to Gompertz]:<math> \gamma=0 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Doubly noncentral t to Noncentral t]:<math> \gamma=0 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Exponential to F]:<math> \alpha=1, X_1/X_2\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html Noncentral F to F]:<math> \delta \to 0\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Exponential to Hyperexponential]:<math> Mixture\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Hyperexponential to Exponential]:<math> \vec \alpha=\alpha \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html IDB to Exponential]:<math>\delta=\kappa \to 0, \alpha=1/ \gamma \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Exponential to Rayleigh]:<math> X^2\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Weibull to Exponential]:<math> \beta=1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Weibull_Distribution.html Exponential to Weibull]:<math> X^{1/\beta}\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html Muth to Exponential]:<math> \alpha=1, \kappa \to 0 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Gompertz_Distribution.html Standard uniform to Gompertz]:<math> \frac{log[1-(log X)(log \kappa)/\delta]}{log \kappa}\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Standard uniform to Exponential Power]:<math> [log(1-log(1-X))/\gamma]^{1/\kappa}\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Error to Laplace]:<math> a=0, b=\alpha/2, c=2\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Error_Distribution.html Laplace to Error]:<math> \alpha_1=\alpha_2 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Log-logistic_Distribution.html Standard uniform to log logistic]:<math> \frac{1}{\lambda}(\frac{1-X}{X})^{1/\kappa} \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Standard uniform to Standard triangular]:<math> X_1-X_2 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/uniform_Distribution.html Standard uniform to uniform]:<math> a+(b-a)X \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Standard uniform to standard power]:<math> X^{1/\beta} \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Standard-uniform_Distribution.html Standard power to standard uniform]:<math> \beta=1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Standard uniform to standard power]:<math> X_(n) \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Standard-power_Distribution.html Minimax to standard power]:<math> \gamma=1 \ </math> | ||
+ | |||
+ | * [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html IDB to Rayleigh]:<math> \delta=2/\alpha, \gamma=0 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Standard-Power_Distribution.html Power to Standard Power]:<math> \alpha=1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Rayleigh_Distribution.html Weibull to Rayleigh]:<math> \beta=2 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Generalized Pareto to Pareto]:<math> \gamma=0, X+\delta \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Standard-triangular_Distribution.html Triangular to standard triangular]:<math> a=-1,b=1,m=0\ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Extreme-value_Distribution.html Weibull to Extreme-value]:<math> logX \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Lomax_Distribution.html Log logistic to lomax]:<math> \kappa=1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Log logistic_Distribution.html Lomax to log logistic]:<math> \kappa=1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Log logistic to logistic]:<math> logX \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:<math> \kappa \to 0 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:<math> \alpha =1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Gaussian_Distribution.html Lévy to Gaussian]:<math> \alpha \to 2</math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Power-series_Distribution.html Modified Power Series to Power series]:<math> u(c)=c \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html BLD1 to Geometric]:<math> g(z)=1-p+pz \ </math> where<math>0<p<1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Borel-Tanner_Distribution.html BLD1 to Borel-Tanner]:<math> g(z)=e^{\lambda (z-1)} , 0<\lambda \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html GLD1 to Binomial]:<math> g(z)=1 \ </math> and <math>f(z)=(q'+p'z)^{n} \ </math> where <math>q'=1-p' \ </math> , <math>0<p'<1 \ </math>, and n is positive integer. | ||
+ | * [http://socr.ucla.edu/htmls/dist/Negative-binomial_Distribution.html GLD1 to Negative binomial]:<math> g(z)=1 \ </math> and <math> f(z)=(q'+p'z)^{n} \ </math> where <math> q'=1+P \ </math> , <math> 0<P \ </math> , and <math> n=-k<0 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Binomial-delta_Distribution.html GLD1 to Binomial-delta]: <math> g(z)=(q+pz)^{m} \ </math> , <math> f(z)=z^{n} \ </math> , <math> mp<1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Binomial-Poisson_Distribution.html GLD1 to Binomial-Poisson]:<math> : g(z)=(q+pz)^{m} \ </math> , <math> f(z)=e^{M(z-1)} \ </math> , <math> mp<1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Binomial-negative-binomial_Distribution.html GLD1 to Binomial-negative-binomial]:<math> g(z)=(q+pz)^{m} \ </math> , <math> f(z)=(Q-Pz)^{-k} \ </math> , <math> mp<1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Poisson-delta_Distribution.html GLD1 to Poisson-delta]: <math> g(z)=e^{\theta (z-1)} \ </math>, <math> f(z)=z^{n} \ </math>, <math> \theta <1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Poisson-Poisson_Distribution.html GLD1 to Poisson-Poisson]: <math> g(z)=e^{\theta (z-1)} , f(z)=e^{M(z-1)} , \theta <1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Poisson-binomial_Distribution.html GLD1 to Poisson-binomial]: <math> g(z)=e^{\theta (z-1)} , f(z)=(q+pz)^{n} , \theta <1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Poisson-negative-binomial_Distribution.html GLD1 to Poisson-negative-binomial]: <math> g(z)=e^{\theta (z-1)} , f(z)=(Q-Pz)^{-k} , \theta <1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Negative-binomial-delta_Distribution.html GLD1 to Negative-binomial-delta]: <math> g(z)=(Q-Pz)^{-k} , f(z)=z^{n} , kP<1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Negative-binomial-Poisson_Distribution.html GLD1 to Negative-binomial-Poisson]: <math> g(z)=(Q-Pz)^{-k} , f(z)=e^{M(z-1)} , kP<1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Negative-binomial-binomial_Distribution.html GLD1 to Negative-binomial-binomial]: <math> g(z)=(Q-Pz)^{-k} , f(z)=(q+pz)^{n} , kP<1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Negative-binomial-negative-binomial_Distribution.html GLD1 to Negative-binomial-negative-binomial]: <math> g(z)=(Q-Pz)^{-k} , f(z)=(Q'-P'z)^{-M} , kP<1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Chi-Square to Poisson]: <math> \left(1-F_{\chi _{2(x+1)}^{2} } (2t/\tau )\right)-\left(1-F_{\chi _{2x}^{2} } (2t/\tau )\right) \ </math> and <math> \lambda =t/\tau \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Positive-Poisson_Distribution.html Left-truncated Poisson to Positive Poisson]: <math> r_{1} =1 \ </math> | ||
+ | * [http://socr.ucla.edu/htmls/dist/Right-truncated-Poisson_Distribution.html Doubly-truncated Poisson to Right-truncated Poisson]: <math> r_{1} =0 \ </math> | ||
+ | |||
+ | |||
+ | |||
<hr> | <hr> | ||
+ | |||
* SOCR Home page: http://www.socr.ucla.edu | * SOCR Home page: http://www.socr.ucla.edu | ||
{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}} | {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=Formulas}} |
Current revision as of 12:57, 18 January 2011
Probability Density Functions (PDFs)
- Standard Normal PDF:
- General Normal PDF:
- Chi-Square PDF:
- Gamma PDF:
- Beta PDF:
- Student's T PDF:
- Poisson PDF:
- Chi PDF:
- Cauchy PDF:
- Exponential PDF:
- F Distribution PDF:
- Bernoulli PMF:
- Binomial PMF:
- Multinomial PMF: , where , , and .
- Negative Binomial PMF:
- Negative-Multinomial Binomial PMF:
- Geometric PMF:
- Erlang PDF:
- Laplace PDF:
- Continuous Uniform PDF:
- Discrete Uniform PMF:
- Logarithmic PDF:
- Logistic PDF:
- Logistic-Exponential PDF:
- Power Function PDF:
- Benford's Law:
- Pareto PDF:
- Non-Central Student T PDF:
- ArcSine PDF:
- Circle PDF:
- U-Quadratic PDF:
- Standard Uniform PDF:
- Zipf:
- Inverse Gamma:
- Fisher-Tippett:
where - Gumbel:
- HyperGeometric:
- Log-Normal:
- Gilbrats:
- Hyperbolic Secant:
- Gompertz:
- Standard Cauchy:
- Rectangular:
- Beta-Binomial:
- Negative Hypergeometric:
- Standard Power:
- Power_Series:
- Zeta:
- Logarithm:
- Beta_Pascal:
- Gamma_Poisson:
- Pascal:
- Polya:
- Normal-Gamma:
- Discrete_Weibull:
- Log Gamma:
- Generalized Gamma:
- Noncentral-Beta:
- Inverse Gausian:
- Noncentral_chi-square:
- Standard Wald:
- Inverted Beta:
- Arctangent:
- Makeham:
- Hypoexponential:
- Doubly Noncentral t:
- Hyperexponential:
- Muth:
- Error:
- Minimax:
- Noncentral F:
- IDB:
- Standard Power:
- Rayleigh:
- Standard Triangular:
- Doubly noncentral F:
- Power:
- Weibull:
- Log-logistic:
- TSP:
- Extreme value:
- Lomax:
- von Mises:
- Generalized Pareto:
- Triangular:
- Kolmogorov-Smirnov:
- Exponential Power:
- Lévy distribution:
- Modified Power Series distributon: where
- Positive binomial distribution: where x = 1,2,...,n
- Basic Lagrangian distribution of the first kind (BLD1): where g(z) is pgf , g(0) is not 0
- General Basic Lagrangian distribution of the first kind (GLD1): Where f(z) and g(z) are pgf , for
- Binomial-delta distribution: for
- Binomial-Poisson distribution: , for
- Binomial-negative-binomial distribution: for
- _Distribution.html Poisson-delta distribution: for
- Poisson-Poisson distribution(also called "Generalized Poisson distribution"): P(X = x) = M(M + θx)x − 1e − (M + θx) / x! for
- Poisson-binomial distribution:
- Poisson-negative-binomial distribution:
- Negative-binomial-delta distribution:
- Negative-binomial-Poisson distribution: , for
- Negative-binomial-binomial distribution: P(X = 0) = qn , for
- Negative-binomial-negative-binomial distribution: for
- Weight binomial distribution:
- Positive Poisson distribution (conditional Poisson distribution): P(X = x) = (eθ − 1) − 1θx / x!,x = 1,2,......
- Left-truncated Poisson distribution:
- Right-truncated Poisson distribution:
- Doubly-truncated Poisson distribution:
- Misrecorded Poisson distribution:
Transformations
- Standard Normal to General Normal Transformation:
- General Normal to Standard Normal Transformation:
- Standard Normal to Chi Transformation:
- Standard Normal to Chi-Square Transformation:
- Gamma to General Normal Transformation:
- Gamma to Exponential Transformation: The special case of is equivalent to exponential Exp(λ).
- Gamma to Beta Transformation: .
- Student's T to Standard Normal Transformation:
- Student's T to Cauchy Transformation:
- Cauchy to General Cauchy Transformation:
- General Cauchy to Cauchy Transformation:
- Fisher's F to Student's T:
- Student's T to Fisher's F: X2
- Bernoulli to Binomial Transformation: (iid)
- Binomial to Bernoulli Transformation:
- Binomial to General Normal Transformation:
- Binomial to Poisson Transformation:
- Multinomial to Binomial Transformation:
- Negative Binomial to Geometric Transformation:
- Erlang to Exponential Transformation:
- Erlang to Chi-Square Transformation:
- Laplace to Exponential Transformation:
- Exponential to Laplace Transformation:
- Beta to Arcsine Transformation:
- Noncentral Student's T to Normal Transformation:
- Noncentral Student's T to Student's T Transformation:
- Standard Uniform to Pareto Transformation:
- Standard Uniform to Benford Transformation:
- Standard Uniform to Exponential Transformation:
- Standard Uniform to Log Logistic Transformation:
- Standard Uniform to Standard Triangular Transformation: X1 − X2
- Standard Uniform to Logistic Exponential Transformation:
- Standard Uniform to Beta Transformation: If X has a standard uniform distribution, has a beta distribution
- Beta to Standard Uniform Transformation: β = γ = 1
- Continuous Uniform to Standard Uniform Transformation:
- Pareto to Exponential:
- Logistic Exponential to Exponential:
- Zipf to Discrete Uniform:
- Discrete Uniform to Rectangular:
- Poisson to Normal:
- Binomial to Poisson:
- Gamma to Inverted Gamma:
- Fisher-Tippett to Gumbel:
- Hypergeometric to Binomial:
- Log-Normal to Normal:
- Normal to Log-Normal:
- Log-Normal to Gibrat's:
- Cauchy to Standard Cauchy:
- Standard Cauchy to Cauchy:
- Standard Cauchy to Hyperbolic Secant:
- Beta to Standard Power:
- Power series to Pascal:
- Gamma Poisson to Pascal:
- Poisson to Gamma Poisson:
- Discrete uniform to Rectangular:
- beta binomial to rectangular:
- beta binomial to negative hypergeometric:
- Zipf to Zeta:
- Power series to Logarithm:
- Power series to Poisson:
- Pascal to Beta pascal:
- pascal to poisson:
- binomial to beta binomial:
- negative hypergeometric to binomial:
- Polya to Binomial:
- Pascal to geometric:
- geometric to pascal:
- discrete weibull to geometric:
- pascal to normal:
- normal to standard normal:
- normal to noncentral_chi-square:
- Normal to Chi-square:
- Beta to Normal:
- Normal to Gamma-normal:
- Standard Normal to Standard Cauchy:
- Inverse Gaussian to Standard normal:
- Noncentral chi-square to Chi-square:
- Gamma to Log gamma:
- Generalized gamma to Log normal:
- Generalized gamma to Gamma:
- Inverse Gaussian to Standard Wald:
- Inverse Gaussian to Chi-square:
- Chi-square to Chi:
- Chi-square to F:
- F to Chi-square:
- Exponential to Chi-square:
- Chi-square to Exponential:
- Chi-square to Erlang:
- Gamma to Chi-square:
- Beta to Standard Uniform:
- Gamma to Erlang:
- Gamma to Inverted Beta:
- Beta to Inverted Beta:
- Cauchy to Arctangent:
- Hypoexponential to Erlang:
- Exponential to Hypoexponential:
- Erlang to Exponential:
- Makeham to Gompertz:
- Doubly noncentral t to Noncentral t:
- Exponential to F:
- Noncentral F to F:
- Exponential to Hyperexponential:
- Hyperexponential to Exponential:
- IDB to Exponential:
- Exponential to Rayleigh:
- Weibull to Exponential:
- Exponential to Weibull:
- Muth to Exponential:
- Standard uniform to Gompertz:
- Standard uniform to Exponential Power:
- Error to Laplace:
- Laplace to Error:
- Standard uniform to log logistic:
- Standard uniform to Standard triangular:
- Standard uniform to uniform:
- Standard uniform to standard power:
- Standard power to standard uniform:
- Standard uniform to standard power:
- Minimax to standard power:
- IDB to Rayleigh:
- Power to Standard Power:
- Weibull to Rayleigh:
- Generalized Pareto to Pareto:
- Triangular to standard triangular:
- Weibull to Extreme-value:
- Log logistic to lomax:
- logistic_Distribution.html Lomax to log logistic:
- Log logistic to logistic:
- TSP to triangular:
- von Mises to Uniform:
- Lévy to Cauchy:
- Lévy to Gaussian:
- Modified Power Series to Power series:
- BLD1 to Geometric: where
- BLD1 to Borel-Tanner:
- GLD1 to Binomial: and where , , and n is positive integer.
- GLD1 to Negative binomial: and where , , and
- GLD1 to Binomial-delta: , ,
- GLD1 to Binomial-Poisson: , ,
- GLD1 to Binomial-negative-binomial: , ,
- GLD1 to Poisson-delta: , ,
- GLD1 to Poisson-Poisson:
- GLD1 to Poisson-binomial:
- GLD1 to Poisson-negative-binomial:
- GLD1 to Negative-binomial-delta:
- GLD1 to Negative-binomial-Poisson:
- GLD1 to Negative-binomial-binomial:
- GLD1 to Negative-binomial-negative-binomial:
- Chi-Square to Poisson: and
- Left-truncated Poisson to Positive Poisson:
- Doubly-truncated Poisson to Right-truncated Poisson:
- SOCR Home page: http://www.socr.ucla.edu
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