Formulas - Revision history

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2020-03-03T19:18:36Z

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ChinPangHo: /* Transformations */

2011-01-18T13:03:10Z

‎Transformations

ChinPangHo: /* Probability Density Functions (PDFs) */

2011-01-18T13:01:06Z

‎Probability Density Functions (PDFs)

ChinPangHo: /* Probability Density Functions (PDFs) */

2011-01-18T12:52:09Z

‎Probability Density Functions (PDFs)

ChinPangHo: /* Probability Density Functions (PDFs) */

2011-01-18T12:49:32Z

‎Probability Density Functions (PDFs)

ChinPangHo: /* Probability Density Functions (PDFs) */

2011-01-18T12:46:07Z

‎Probability Density Functions (PDFs)

ChinPangHo: /* Transformations */

2011-01-11T13:56:39Z

‎Transformations

ChinPangHo: /* Transformations */

2011-01-11T13:56:15Z

‎Transformations

ChinPangHo: /* Transformations */

2011-01-11T13:45:04Z

‎Transformations

ChinPangHo: /* Transformations */

2011-01-11T13:44:31Z

‎Transformations

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

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	* [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:<math> n=2 \ </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/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>

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 * [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)=\left\binom{n}{x}\right{p^{x} q^{n-x} \over (1-q^{n} )} </math>          where    <math>  x=1,2,...,n </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) ,

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 * [http://socr.ucla.edu/htmls/dist/Positive-binomial_Distribution.html Positive binomial distribution]: <math> P(X=x)=\left\binom{n}{x}\right{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)} \cr ,
+* [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>

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 * [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)=\left\binom{n}{x}\right{p^{x} q^{n-x} \over (1-q^{n} )} </math>        where    <math>  x=1,2,...,n </math>
+* [http://socr.ucla.edu/htmls/dist/Positive-binomial_Distribution.html Positive binomial distribution]: <math> P(X=x)=\left\binom{n}{x}\right{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)} \cr ,

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 * [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>
 ==Transformations==

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 * [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/Poisson-Poisson_Distribution.html GLD1 to Poisson-Poisson]: <math> g(z)=e^{\theta (z-1)}   ,  f(z)=e^{M(z-1)}   ,  \theta <1 \ </math>