Formulas
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==Probability Density Functions (PDFs)== | ==Probability Density Functions (PDFs)== | ||
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* [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 | + | * [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 | + | * [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 | + | * [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 | + | * [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> | ||
==Transformations== | ==Transformations== |
Revision as of 17:24, 4 November 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:
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:
- SOCR Home page: http://www.socr.ucla.edu
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