Formulas

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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\[ f(x)=\begin{cases} \frac{2(x-a)}{(b-a)(m-a)}, a<x<m \\ \frac{2(b-x)}{(b-a)(b-m)}, m \le x<b \end{cases}. a<m<b>0 \!\]
  • 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\[ P(X=0)=\omega +(1-\omega )e^{-\theta }, P(X=x)=(1-\omega ){e^{-\theta } \theta ^{x} \over x!} , x\ge 1\]

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