Formulas

From Socr

Revision as of 19:41, 22 April 2010 by John guojun (Talk | contribs)

(diff) ← Older revision | Current revision (diff) | Newer revision → (diff)

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 $x_1+x_2+\cdots+x_k=n$ , $p_1+p_2+\cdots+p_k=1$ , and $0 \le x_i \le n, 0 \le p_i \le 1$ .
Negative Binomial PMF: $\begin{pmatrix} k + r - 1 \\ k \end{pmatrix} p^r(1-p)^k$
Negative-Multinomial Binomial PMF: $P(k_o, \cdots, k_r) = \Gamma(k_o + \sum_{i=1}^r{k_i}) \frac{p_o^{k_o}}{\Gamma(k_o)} \prod_{i=1}^r{\frac{p_i^{k_i}}{k_i!}}$
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: $f(x; 0,1) = \frac{1}{\pi (1 + x^2)}. \!$
Rectangular: $f(x)=\frac{1}{n+1}.(x=0,1,...,n)\!$
Beta-Binomial: $f(x)=\frac{\Gamma(x+a)\Gamma(n-x+b)\Gamma(a+b)\Gamma(n+2)}{(n+1)\Gamma(a+b+n)\Gamma(a)\Gamma(b)\Gamma(x+1)\Gamma(n-x+1)}.(x=0,1,...,n)\!$
Negative Hypergeometric: $f(x)=\frac{\begin{pmatrix} n_1+x-1 \\ x \end{pmatrix} \begin{pmatrix} n_3-n_1+n_2-x-1 \\ n_2-x \end{pmatrix}}{\begin{pmatrix} n_3+n_2-1 \\ n_2 \end{pmatrix}}. (x=max(0,n_1+n_2-n_3),...,n_2)\!$
Standard Power: $f(x; \beta) = \beta x^{\beta - 1} \!$
Power_Series: $f(x; c; A(c)) = a(x) c^x / A(c). (x=(0,1,...), c>0, A(c)=\sum_{x}a(x) c^x) \!$
Zeta: $f(x)=\frac{1}{x^a \sum_{i=1}^{\infty}(\frac{1}{i})^a}. (x=1,2,...) \!$
Logarithm: $f(x)=\frac{-(1-c)^x}{x\log c}. (x=1,2,..., 0<c<1) \!$
Beta_Pascal: $f(x; a, b, n) = \binom{n-1+x}{x} \frac{B(n+a, b+x)}{B(a,b)}. (x=(0,1,...); a+b=n) \!$
Gamma_Poisson: $f(x; \alpha, \beta) = \frac{\Gamma(x+\beta) \alpha^x}{\Gamma(\beta) (1+\alpha)^{\beta+x} x!}.(x=(0,1,...); \alpha>0; \beta>0) \!$
Pascal: $f(x; p, n) = \binom{n-1+x}{x} p^n (1-p)^x. (x=(0,1,...,n); 0 \leq p \leq 1)\!$
Polya: $f(x; n, p, \beta) = \binom{n}{x} \frac{\prod_{j=0}^{x-1}(p+j\beta) \prod_{k=0}^{n-x-1}(1-p+k\beta)}{\prod_{i=0}^{n-1}(1+i\beta)}. (x=\{0,1,...,n\}) \!$
Normal-Gamma: $f(x, \tau; \mu, \lambda,\alpha,\beta) = \frac{\beta^\alpha \sqrt(\lambda)}{\Gamma(\alpha) \sqrt(2 \pi)} \tau^{\alpha-1/2} exp(-\beta \tau) exp(-\frac{\lambda \tau (x-\mu)^2}{2}).(\tau>0) \!$
Discrete_Weibull: Failed to parse (PNG conversion failed;

check for correct installation of latex, dvips, gs, and convert): f(x; p, \beta) = (1-p)^x^\beta-(1-p)^(x+1)^\beta. (x=\{0,1,...\}) \!

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: $\begin{vmatrix}\mu = np \\ n \rightarrow \infty \end{vmatrix}$
Multinomial to Binomial Transformation: $\begin{vmatrix} k=2 \end{vmatrix}$
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: $X 1 - X 2$
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} \$
Beta to Standard Power: $\alpha=\beta, \beta=1 \$

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

Translate this page:

(default)	Deutsch	Español	Français	Italiano	Português	日本語	България	الامارات العربية المتحدة	Suomi	इस भाषा में	Norge
한국어	中文	繁体中文	Русский	Nederlands	Ελληνικά	Hrvatska	Česká republika	Danmark	Polska	România	Sverige

Formulas

From Socr

Probability Density Functions (PDFs)

Transformations

Views

Personal tools

Navigation

Search

Toolbox