# Formulas

(Difference between revisions)
 Revision as of 12:38, 1 December 2008 (view source)Jczhang (Talk | contribs)m (→Transformations)← Older edit Revision as of 22:24, 23 October 2009 (view source)IvoDinov (Talk | contribs) (→Probability Density Functions (PDFs): added Multinomial)Newer edit → Line 17: Line 17: \mbox{0 otherwise} \end{cases} [/itex] \mbox{0 otherwise} \end{cases} [/itex] * [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: $\begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}$ * [http://socr.ucla.edu/htmls/dist/Binomial_Distribution.html Binomial] PMF: $\begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}$ + *  [http://socr.ucla.edu/htmls/dist/Multinomiall_Distribution.html Multinomial] PMF: $f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}$, where $\forall x_1+x_2+\cdots+x_k=n$, and $\forall p_1+p_2+\cdots+p_k=1$. * [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: $\begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k$ * [http://socr.ucla.edu/htmls/dist/NegativeBinomial_Distribution.html Negative Binomial] PMF: $\begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k$ * [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: $\begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p$ * [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html Geometric] PMF: $\begin{pmatrix} 1-p \end{pmatrix} ^{k-1}p$

## 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: $\begin{pmatrix} n \\ k \end{pmatrix} p^k (1-p)^{n-k}$
• Multinomial PMF: $f(x_1, x_2, \cdots, x_k)={n\choose x_1,x_2,\cdots, x_k}p_1^{x_1}p_2^{x_2}\cdots p_k^{x_k}$, where $\forall x_1+x_2+\cdots+x_k=n$, and $\forall p_1+p_2+\cdots+p_k=1$.
• 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: $f(t)=\frac{\nu^{\nu/2}e^{-\nu\mu^2/2(t^2+\nu)}} {\sqrt{\pi}\Gamma(\nu/2)2^{(\nu-1)/2}(t^2+\nu)^{(\nu+1)/2}} \times\int\limits_0^\infty x^\nu\exp\left[-\frac{1}{2}\left(x-\frac{\mu t}{\sqrt{t^2+\nu}}\right)^2\right]dx$
• ArcSine PDF: $f(x) = \frac{1}{\pi \sqrt{x(1-x)}}$
• Circle PDF: $f(x)={2\sqrt{r^2 - x^2}\over \pi r^2 }, \forall x \in [-r , r]$
• U-Quadratic PDF: $\alpha \left ( x - \beta \right )^2$
• Standard Uniform PDF: $U(0,1) = f(x) = \begin{cases} {1} \mbox{ for } 0 \le x \le 1 \\ 0 \mbox{ for } x < 0 \mbox{ or } x > 1 \end{cases}$
• Zipf: $\frac{1/(k+q)^s}{H_{N,s}}$
• Inverse Gamma: $\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)$
• Fisher-Tippett: $\frac{z\,e^{-z}}{\beta}\!$
where $z = e^{-\frac{x-\mu}{\beta}}\!$
• Gumbel: $f(x) = e^{-x} e^{-e^{-x}}.$
• HyperGeometric: ${{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}$
• Log-Normal: $\frac{1}{x\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)-\mu\right)^2}{2\sigma^2}\right]$
• Gilbrats: $\frac{1}{\sigma\sqrt{2\pi}}\exp\left[-\frac{\left(\ln(x)\right)^2}{2\sigma^2}\right]$
• Hyperbolic Secant: $\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!$
• Gompertz: $b e^{-bx} e^{-\eta e^{-bx}}\left[1 + \eta\left(1 - e^{-bx}\right)\right]$
• Standard Cauchy: $f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!$

## 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 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 \$
• Fisher's F to Student's T: $\sqrt X$
• Student's T to Fisher's F: X2
• 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 \$
• Beta to Arcsine Transformation: $\alpha = \beta = \frac{1}{2}$
• Noncentral Student's T to Normal Transformation: $Z=\lim_{\nu\to\infty}T$
• Noncentral Student's T to Student's T Transformation: $\mu = 0 \$
• Standard Uniform to Pareto Transformation: $\lambda X ^{-1/K} \$
• Standard Uniform to Benford Transformation: $10^X \$
• Standard Uniform to Exponential Transformation: $n(1-X_{(n)}), n -> \infty$
• Standard Uniform to Log Logistic Transformation: $\frac{1}{\lambda}(\frac{1-X}{X})^{1/k}$
• Standard Uniform to Standard Triangular Transformation: X1X2
• Standard Uniform to Logistic Exponential Transformation: $\frac{log[1+(\frac{X}{1-X})^{1/K}]}{\lambda}$
• Standard Uniform to Beta Transformation: If X has a standard uniform distribution, $Y = 1 - X^{1/n} \$ has a beta distribution
• Beta to Standard Uniform Transformation: β = γ = 1
• Continuous Uniform to Standard Uniform Transformation: $a = 0, b = 1 \$
• Pareto to Exponential: $log(X/\lambda) \$
• Logistic Exponential to Exponential: $\beta = 1 \$
• Zipf to Discrete Uniform: $a = 0, a = 1, b = n \$
• Discrete Uniform to Rectangular: $a = 0, b = n \$
• Poisson to Normal: $\sigma ^2 = \mu , \mu \to \infty$
• Binomial to Poisson: $\mu = np, \mu \to \infty$
• Gamma to Inverted Gamma: $\frac{1}{X}$
• Fisher-Tippett to Gumbel: $\mu = 0, \beta = 1 \$
• Hypergeometric to Binomial: $p = \frac{n_1}{n_3}, n = n_2, n_3, n \to \infty \$
• Log-Normal to Normal: $log(X) \$
• Normal to Log-Normal: $e^X \$
• Log-Normal to Gibrat's: $\mu = 0, x = 1 \$
• Cauchy to Standard Cauchy: $\gamma = 1, x_0 = 0 \$
• Standard Cauchy to Cauchy: $x_0 + \gamma X \$
• Standard Cauchy to Hyperbolic Secant: $\frac{log|x|}{\pi} \$