Formulas
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* [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/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> |
Revision as of 05:29, 29 October 2009
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:
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:
- SOCR Home page: http://www.socr.ucla.edu
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