Formulas

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\[ q = (1 - p) \] for \( k = 0, p \) for \( k = 1 \)
• 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\[ \frac {1}{2b} \exp (- \frac{|x-\mu|}{b}) \]

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 \(Exp(\lambda)\).
• 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\[\begin{pmatrix} \begin{vmatrix} X \end{vmatrix} \\ \alpha_1 = \alpha_2 \end{pmatrix}\]

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