# Formulas

(Difference between revisions)
 Revision as of 20:45, 24 April 2008 (view source)IvoDinov (Talk | contribs) (→Transformations)← Older edit Revision as of 20:50, 24 April 2008 (view source)IvoDinov (Talk | contribs) (→Transformations)Newer edit → Line 25: Line 25: * [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: $n\longrightarrow\infty$ * [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Standard Normal Transformation]: $n\longrightarrow\infty$ * [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: $n=1 \$ * [http://en.wikipedia.org/wiki/Student's_t_distribution Student's T to Cauchy Transformation]: $n=1 \$ - + * [http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy to General Cauchy Transformation]: $a + \alpha\times X$ + * [http://en.wikipedia.org/wiki/Cauchy_distribution General Cauchy to Cauchy Transformation]: $a=0; \alpha=1 \$

## Revision as of 20:50, 24 April 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: $\lambda e^{-\lambda x},\; x \ge 0$

## 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 Exp(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 \$