Formulas

From SOCR
Jump to: navigation, search

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\[ f(x; p, \beta) = (1-p)^{x^\beta}-(1-p)^{(x+1)^\beta}. (x=\{0,1,...\}) \!\]
  • Log Gamma\[ f(x)=[1/ \alpha^\beta \Gamma(\beta)]e^{\beta x}e^{-e^x/a}. (-\infty<x<\infty) \!\]
  • Generalized Gamma\[ f(x)=\frac{\gamma}{\alpha^{\gamma \beta}\Gamma(\beta)}x^{\gamma \beta-1}e^{-(x/\alpha)^\gamma}. (x>0) \!\]
  • Noncentral-Beta\[ f(x; \beta, \gamma, \delta) = \sum_{i=0}^{\infty}\frac{\Gamma(i+\beta+\gamma)}{\Gamma(\gamma) \Gamma(i+\beta)} \frac{exp(-\delta/2)}{i!} (\delta/2)^i x^{i+\beta-1} (1-x)^{\gamma-1}. (0 \leq x \leq 1). \!\]
  • Inverse Gausian\[ f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2\mu^2 x}(x-\mu)^2}. (x>0) \!\]
  • Noncentral_chi-square\[ f(x; n,\delta) = f(x; n,\delta) = \sum_{k=0}^{\infty}\frac{exp(-\delta/2) (\delta/2)^k}{k!}\frac{exp(-x/2) x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma(\frac{n+2k}{2})}. \!\]
  • Standard Wald\[ f(x)=\sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda}{2x}(x-1)^2}. (x>0) \!\]
  • Inverted Beta\[ f(x)=\frac{x^{\beta-1}(1+x)^{-\beta-\gamma}}{B(\beta,\gamma)}. (x>0, \beta>1, \gamma>1) \!\]
  • Arctangent\[ f(x; \lambda, \phi)= \frac{\lambda}{[arctan(\lambda \phi)+\pi/2][1+\lambda^2 (x - \phi)^2]} (x \geq 0, -\infty < \lambda < \infty) \!\]
  • Makeham\[ f(x) = (\gamma + \delta\kappa^x)exp(-\gamma x-\frac{\delta (\kappa^x-1)}{log(\kappa)}). x>0 \!\]
  • Hypoexponential\[ f(x) = \sum_{i=1}^{n}(1/\alpha_i)exp(-x/\alpha_i)(\prod_{j=1,j\neq i}^{n}\frac{\alpha_i}{\alpha_i-\alpha_j}). x>0 \!\]
  • Doubly Noncentral t\[ \!\]
  • Hyperexponential\[ f(x) = \sum_{i=1}^{n}\frac{p_i}{\alpha_i}e^{-x/\alpha_i}. x>0 \!\]
  • Muth\[ f(x) = (e^{\kappa x}-\kappa)e^{-(1/\kappa)e^{\kappa x}+\kappa x+1/\kappa}. x>0 \!\]
  • Error\[ f(x) = \frac{exp[-(|x-a|/b)^{2/c}/2]}{b 2^{c/2+1}\Gamma(1+c/2)}. -\infty < x < \infty \!\]
  • Minimax\[ f(x) = \beta\gamma x^{\beta-1}(1-x^\beta)^{\gamma-1}. 0<x<1 \!\]
  • Noncentral F\[ f(x) = \sum_{i=0}^{\infty}\frac{\Gamma(\frac{2i+n_1+n_2}{2})(n_1/n_2)^{(2i+n_1)/2}x^{(2i+n_1-2)/2}e^{-\delta/2}(\delta/2)^i}{\Gamma(n_2/2)\Gamma(\frac{2i+n_1}{2})i!(1+\frac{n_1}{n_2}x)^{(2i+n_1+n_2)/2}}. x>0 \!\]
  • IDB\[ f(x) = \frac{(1+\kappa x)\delta x+\gamma}{(1+\kappa x)^{\gamma/\kappa+1}}e^{-\delta x^2/2}. x>0 \!\]
  • Standard Power\[ f(x) = \beta x^{\beta-1}. 0<x<1 \!\]
  • Rayleigh\[ f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!\]
  • Standard Triangular\[ f(x) = \begin{cases} x+1, -1<x<0 \\ 1 - x, 0 \leq x<1 \end{cases} \!\]
  • Doubly noncentral F\[ f(x)= \sum_{j=0}^{\infty}\sum_{k=0}^{\infty}[\frac{e^{-\delta/2}(\frac{1}{2}\delta)^j}{j!}][\frac{e^{-\gamma/2}(\frac{1}{2}\gamma)^k}{k!}]\times n_1^{(n_1/2)+j}n_2^{(n_2/2)+k}x^{(n_1/2)+j-1}\times (n_2+n_1 x)^{-\frac{1}{2}(n_1+n_2)-j-k}\times [B(\frac{1}{2}n_1+j,\frac{1}{2}n_2+k)]^{-1}. x>0 \!\]
  • Power\[ f(x)=\frac{\beta x^{\beta-1}}{\alpha^\beta}. 0<x<\alpha \!\]
  • Weibull\[ f(x)=(\beta/\alpha)x^{\beta-1}exp[-(1/\alpha)x^\beta]. x>0 \!\]
  • Log-logistic\[ f(x)=\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{[1+(\lambda x)^\kappa]^2}. x>0 \!\]
  • TSP\[ f(x) = \begin{cases} \frac{n}{b-a}(\frac{x-a}{m-a})^{n-1}, a<x\le m \\ \frac{n}{b-a}(\frac{b-x}{b-m})^{n-1}, m\le x<b \end{cases} \!\]
  • Extreme value\[ f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\infty<x<\infty \!\]
  • Lomax\[ f(x)=\frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}. x>0 \!\]
  • von Mises\[ f(x)=\frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)}. 0<x<2\pi, 0<\mu<2\pi) \!\]
  • Generalized Pareto\[ f(x)=(\gamma+\frac{\kappa}{x+\delta})(1+x/\delta)^{-\kappa}e^{-\gamma x}. x>0 \!\]
  • Triangular\[ f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\ \frac{2(b-x)}{(b-a)(b-m)}, m \le x<b \end{cases}. a<m0 \!\]
  • Lévy distribution\[ L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2 \]
  • Modified Power Series distributon\[ P(X=x)={a(x)\left\{u(c)\right\}^{x} \over A(c)} \] where \( A(c)=\sum _{x}a(x)\left\{u(c)\right\}^{x} ,a(x)\ge 0 \)
  • Positive binomial distribution\[ P(X=x)=\binom{n}{x}{p^{x} q^{n-x} \over (1-q^{n} )} \] where \( x=1,2,...,n \)
  • Basic Lagrangian distribution of the first kind (BLD1)\[ P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } (g(z))^{x} \right]_{z=0} \] where \( g(z) \) is pgf , \( g(0) \) is not 0
  • General Basic Lagrangian distribution of the first kind (GLD1)\[ P(X=0)=f(0) , P(X=x)={1\over x!} \left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} , x>0\] Where f(z) and g(z) are pgf , \(\left[{\partial ^{x-1} \over \partial z^{x-1} } \left\{(g(z))^{x} {\partial f(z)\over \partial z} \right\}\right]_{z=0} >0\) for \(x\ge 1\)
  • Binomial-delta distribution\[ P(X=x)={n\over x}\binom[[:Template:Mx]]{x-n}p^{x-n} q^{n+mx-x} \] for \(x\ge n\)
  • Binomial-Poisson distribution\[ P(X=x)=e^{-M} {(Mq^{m} )^{x} \over x!} {}_{2} F{}_{0} [1-x,-mx;{p\over Mq} ] \] , for \(x\ge 0 \)
  • Binomial-negative-binomial distribution\[ P(X=x)={\Gamma (k+x)\over x!\Gamma (x)} Q^{-k} \left({Pq^{m} \over Q} \right)^{x} {}_{2} F_{1} [1-x,-mx;1-x-k;{-pQ\over qP} ] \] for \(x\ge 0\)
  • _Distribution.html Poisson-delta distribution\[ P(X=x)={n\over x} {e^{-\theta x} (\theta x)^{x-n} \over (x-n)} \] for \(x\ge n \)
  • Poisson-Poisson distribution(also called "Generalized Poisson distribution")\[ P(X=x)=M(M+\theta x)^{x-1} e^{-(M+\theta x)} /x! \] for \(x\ge 0 \)
  • Poisson-binomial distribution\[ P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} npq^{n-1} {}_{2} F_{0} [1-x,1-n;{p\over \theta qx} ] , x\ge 1\]
  • Poisson-negative-binomial distribution\[ P(X=x)={(\theta x)^{x-1} \over x!} e^{-\theta x} kPQ^{-k-1} {}_{2} F_{0} [1-x,1+k;{-P\over \theta Qx} ] , x\ge 1\]
  • Negative-binomial-delta distribution\[ P(X=x)={n\over x} {\Gamma (kx+x-1)\over (x-n)!\Gamma (kx)} \left({P\over Q} \right)^{x-n} Q^{-kx} , x\ge n \]
  • Negative-binomial-Poisson distribution\[ P(X=x)={e^{-M} M^{x} \over x!} Q^{-kx} {}_{2} F_{0} [1-x,kx;-;{-P\over MQ} ] \] , for \(x\ge 0\)
  • Negative-binomial-binomial distribution\[ P(X=0)=q^{n} \] , \(P(X=x)=npq^{n-1} {\Gamma (kx+x-1)\over x!\Gamma (kx)} \left({P\over Q} \right)^{x-1} Q^{-kx} {}_{2} F_{1} [1-x,1-n;2-x-kx;{-pQ\over Pq} ] \) for \(x\ge 1\)
  • Negative-binomial-negative-binomial distribution\[ P(X=x)=(Q')^{-M} \left({P'\over Q'Q^{k} } \right)^{x} {\Gamma (M+x)\over x!\Gamma (M)} {}_{2} F_{1} [1-x,kx;1-M-x;{PQ'\over P'Q} ] \] for \(x\ge 1\)
  • Weight binomial distribution\[ P(X=x)=w(x)p_{x} /\sum _{x}^{}w(x)p_{x}\]
  • Positive Poisson distribution (conditional Poisson distribution)\[ P(X=x)=(e^{\theta } -1)^{-1} \theta ^{x} /x! , x=1,2,......\]
  • Left-truncated Poisson distribution\[ P(X=x)={e^{-\theta } \theta ^{x} \over x!} \left[1-e^{-\theta } \sum _{j=0}^{r_{1} -1}{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...\]
  • Right-truncated Poisson distribution\[ P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=0}^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=0,1,...,r_{2}\]
  • Doubly-truncated Poisson distribution\[ P(X=x)={\theta ^{x} \over x!} \left[\sum _{j=r_{1} }^{r_{2} }{\theta ^{j} \over j!} \right]^{-1} , x=r_{1} ,r_{1} +1,...,r_{2} , 0<r_{1} <r_{2}\]
  • Misrecorded Poisson distribution\[ P(X=0)=\omega +(1-\omega )e^{-\theta }, P(X=x)=(1-\omega ){e^{-\theta } \theta ^{x} \over x!} , x\ge 1\]

Transformations






Translate this page:

(default)
Uk flag.gif

Deutsch
De flag.gif

Español
Es flag.gif

Français
Fr flag.gif

Italiano
It flag.gif

Português
Pt flag.gif

日本語
Jp flag.gif

България
Bg flag.gif

الامارات العربية المتحدة
Ae flag.gif

Suomi
Fi flag.gif

इस भाषा में
In flag.gif

Norge
No flag.png

한국어
Kr flag.gif

中文
Cn flag.gif

繁体中文
Cn flag.gif

Русский
Ru flag.gif

Nederlands
Nl flag.gif

Ελληνικά
Gr flag.gif

Hrvatska
Hr flag.gif

Česká republika
Cz flag.gif

Danmark
Dk flag.gif

Polska
Pl flag.png

România
Ro flag.png

Sverige
Se flag.gif