Formulas
From Socr
(Difference between revisions)
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\mbox{1 - p if k = 0,} \\ | \mbox{1 - p if k = 0,} \\ | ||
\mbox{0 otherwise} \end{cases} </math> | \mbox{0 otherwise} \end{cases} </math> | ||
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* [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/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> | ||
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* [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/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/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> | ||
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* [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/ZipfMandelbrot_Distribution.html Zipf]: <math>\frac{1/(k+q)^s}{H_{N,s}}</math> | ||
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* [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/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/FisherTippett_Distribution.html Fisher-Tippett]: <math>\frac{z\,e^{-z}}{\beta}\!</math><br /> where <math>z = e^{-\frac{x-\mu}{\beta}}\!</math> | ||
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* [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/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/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> | ||
==Transformations== | ==Transformations== |
Revision as of 10:49, 1 December 2008
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:
- Negative 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:
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:
- 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:
- SOCR Home page: http://www.socr.ucla.edu
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