Formulas

From Socr

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Revision as of 08:18, 28 October 2008

This SOCR Wiki page contains a number of formulas, mathematical expressions and symbolic representations that are used in varieties of SOCR resources. Usage is defined as a reference by image, text, TeX, URL, etc. For instance the SOCR Distributome project uses these formulas to represent PDFs, CDFs, transformations, etc.

Probability Density Functions (PDFs)

Standard Normal PDF: $f(x)= {e^{-x^2} \over \sqrt{2 \pi}}$
General Normal PDF: $f(x)= {e^{{-(x-\mu)^2} \over 2\sigma^2} \over \sqrt{2 \pi\sigma^2}}$
Chi-Square PDF: $\frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}\,$
Gamma PDF: $x^{k-1} \frac{\exp{\left(-x/\theta\right)}}{\Gamma(k)\,\theta^k}\,\!$
Beta PDF: $\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1}\!$
Student's T PDF: $\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{x^2}{\nu} \right)^{-(\frac{\nu+1}{2})}\!$
Poisson PDF: $\frac{e^{-\lambda} \lambda^k}{k!}\!$
Chi PDF: $\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}$
Cauchy PDF: $\frac{1}{\pi\gamma \left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}$
Exponential PDF: $\lambda e^{-\lambda x},\; x \ge 0$
F Distribution PDF: $\frac {(\frac {d_1 x}{d_1 x + d_2})^{ d_1/2} ( 1 - \frac {d_1 x} {d_1 x + d2}) ^ {d_2/2}} { xB(d_1/2 , d_2/2) }$
Bernoulli PMF: $f(k;p) \begin{cases} \mbox{p if k = 1,} \\ \mbox{1 - p if k = 0,} \\ \mbox{0 otherwise} \end{cases}$

Binomial PMF: $\begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}$
Negative Binomial PMF: $\begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k$
Geometric PMF: $\begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p$
Erlang PDF: $\frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!}$
Laplace PDF: $\frac {1}{2b} \exp (- \frac{|x-\mu|}{b})$
Continuous Uniform Distribution PDF: $f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases}$
Discrete Uniform Distribution PMF: $f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases}$
Logarithmic Distribution PDF: $f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k}$
Logistic Distribution PDF: $f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2}$

Transformations

Standard Normal to General Normal Transformation: $\mu+\sigma\times X$
General Normal to Standard Normal Transformation: $X-\mu \over \sigma$
Standard Normal to Chi Transformation: $|\ X |$
Standard Normal to Chi-Square Transformation: $\sum_{k=1}^{\nu} X_k^2$
Gamma to General Normal Transformation: $\mu=\alpha\times\beta;\sigma^2=\alpha^2\times\beta;\beta\longrightarrow\infty$
Gamma to Exponential Transformation: The special case of ${\Gamma}(k=1, \theta=1/\lambda)\,$ is equivalent to exponential $E x p (λ)$ .
Gamma to Beta Transformation: $X_1 \over X_1 + X_2$ .
Student's T to Standard Normal Transformation: $n\longrightarrow\infty$
Student's T to Cauchy Transformation: $n=1 \$
Cauchy to General Cauchy Transformation: $a + \alpha\times X$
General Cauchy to Cauchy Transformation: $a=0; \alpha=1 \$
Fisher's F to Student's T: $\sqrt X$
Student's T to Fisher's F: $X 2$
Bernoulli to Binomial Transformation: $\sum X_i$ (iid)
Binomial to Bernoulli Transformation: $\begin{pmatrix} n = 1 \end{pmatrix}$
Binomial to General Normal Transformation: $\begin{vmatrix} \mu = np \\ \sigma^2 = np(1-p) \\n \rightarrow \infty \end{vmatrix}$
Binomial to Poisson Transformation: $\begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix}$
Negative Binomial to Geometric Transformation: $\begin{pmatrix} r = 1 \end{pmatrix}$
Erlang to Exponential Transformation: $\begin{pmatrix} k = 1 \end{pmatrix}$
Erlang to Chi-Square Transformation: $\begin{pmatrix} \alpha = 2 \end{pmatrix}$
Laplace to Exponential Transformation: $\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}$
Exponential to Laplace Transformation: $x_1 - x_2 \$

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

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 * [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform Distribution] 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 Distribution] PDF: <math> f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k}  </math>
-* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic Distribution] PDF:
+* [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic Distribution] PDF: <math> f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2} </math>
 ==Transformations==

Formulas

From Socr

Revision as of 08:18, 28 October 2008

Probability Density Functions (PDFs)

Transformations

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