# Formulas

(Difference between revisions)
 Revision as of 08:18, 28 October 2008 (view source)Jczhang (Talk | contribs)m (→Probability Density Functions (PDFs))← Older edit Revision as of 17:24, 4 November 2008 (view source)Jenny (Talk | contribs) Newer edit → Line 1: Line 1: - This [[Main_Page | 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 [http://socr.ucla.edu/htmls/SOCR_Distributome.html SOCR Distributome project] uses these formulas to represent PDFs, CDFs, transformations, etc. - ==Probability Density Functions (PDFs)== ==Probability Density Functions (PDFs)== Line 24: Line 22: * [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: $\frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!}$ * [http://socr.ucla.edu/htmls/dist/Erlang_Distribution.html Erlang] PDF: $\frac {\lambda x^{k-1}e^{-\lambda x}} {(k-1)!}$ * [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: $\frac {1}{2b} \exp (- \frac{|x-\mu|}{b})$ * [http://socr.ucla.edu/htmls/dist/Laplace_Distribution.html Laplace] PDF: $\frac {1}{2b} \exp (- \frac{|x-\mu|}{b})$ - * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform Distribution] PDF: $f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases}$ + * [http://socr.ucla.edu/htmls/dist/ContinuousUniform_Distribution.html Continuous Uniform] PDF: $f(x) = \begin{cases} \frac{1}{b-a} \mbox{ for } a \le x \le b \\ 0 \mbox{ for } x < a \mbox{ or } x > b \end{cases}$ - * [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform Distribution] PMF: $f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases}$ + * [http://socr.ucla.edu/htmls/dist/DiscreteUniform_Distribution.html Discrete Uniform] PMF: $f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases}$ - * [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic Distribution] PDF: $f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k}$ + * [http://socr.ucla.edu/htmls/dist/LogarithmicSeries_Distribution.html Logarithmic] PDF: $f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k}$ - * [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic Distribution] PDF: $f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2}$ + * [http://socr.ucla.edu/htmls/dist/Logistic_Distribution.html Logistic] PDF: $f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2}$ + * [http://socr.ucla.edu/htmls/dist/LogisticExponential_Distribution.html Logistic-Exponential] PDF: $f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0$ + * [http://socr.ucla.edu/htmls/dist/PowerFunction_Distribution.html Power Function] PDF: $f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha}$ + * [http://socr.ucla.edu/htmls/dist/Benford_Distribution.html Benford's Law]: $P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d})$ + * [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html Pareto] PDF: $\frac {kx^k_m} {x^{k+1}}$ + * [http://socr.ucla.edu/htmls/dist/StudentT_Distribution.html Non-Central Student T] PDF: $f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx$ ==Transformations== ==Transformations==

## 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 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 PMF: $f(x) = \begin{cases} 1/n \mbox{ for } a \le x \le b, \\ 0 \mbox{ otherwise} \end{cases}$
• Logarithmic PDF: $f(k) = \frac{-1}{ln(1-p)} \frac{p^k}{k}$
• Logistic PDF: $f(x;u,s) = \frac{e^{-(x-\mu)/s}} {s(1+e^{-(x-\mu)/s})^2}$
• Logistic-Exponential PDF: $f(x;\beta) = \frac { \beta e^x(e^x - 1)^{\beta-1}} {(1+(e^x-1)^\beta))^2} \mbox{ }\mbox{ }x, \beta > 0$
• Power Function PDF: $f(x) = \frac {\alpha(x-a)^{\alpha-1}} {(b-a)^\alpha}$
• Benford's Law: $P(d) = \log_b(d + 1)- \log_b(d) = \log_b(\frac{d + 1}{d})$
• Pareto PDF: $\frac {kx^k_m} {x^{k+1}}$
• Non-Central Student T PDF: $f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx$