# Formulas

(Difference between revisions)
 Revision as of 12:46, 18 January 2011 (view source) (→Probability Density Functions (PDFs))← Older edit Current revision as of 12:57, 18 January 2011 (view source) (→Transformations) (One intermediate revision not shown) Line 97: Line 97: * [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: $L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2$ * [http://socr.ucla.edu/htmls/dist/Lévy_Distribution.html Lévy distribution]: $L_{\alpha ,\gamma } (y)={1\over \pi } \int _{0}^{\infty }e^{-\gamma q^{\alpha } } \cos (qy) dq , y\in {\rm R} , \gamma >0 , 0<\alpha <2$ * [http://socr.ucla.edu/htmls/dist/Modified-Power-Series_Distribution.html Modified Power Series distributon]: $P(X=x)={a(x)\left\{u(c)\right\}^{x} \over A(c)}$  where  $A(c)=\sum _{x}a(x)\left\{u(c)\right\}^{x} ,a(x)\ge 0$ * [http://socr.ucla.edu/htmls/dist/Modified-Power-Series_Distribution.html Modified Power Series distributon]: $P(X=x)={a(x)\left\{u(c)\right\}^{x} \over A(c)}$  where  $A(c)=\sum _{x}a(x)\left\{u(c)\right\}^{x} ,a(x)\ge 0$ - * [http://socr.ucla.edu/htmls/dist/Positive-binomial_Distribution.html Positive binomial distribution]: $P(X=x)=\left\binom{n}{x}\right{p^{x} q^{n-x} \over (1-q^{n} )}$          where    $x=1,2,...,n$ + * [http://socr.ucla.edu/htmls/dist/Positive-binomial_Distribution.html Positive binomial distribution]: $P(X=x)=\binom{n}{x}{p^{x} q^{n-x} \over (1-q^{n} )}$          where    $x=1,2,...,n$ * [http://socr.ucla.edu/htmls/dist/Basic-Lagrangian-distribution-of-the-first-kind.html 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 * [http://socr.ucla.edu/htmls/dist/Basic-Lagrangian-distribution-of-the-first-kind.html 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 * [http://socr.ucla.edu/htmls/dist/General-Basic-Lagrangian-distribution-of-the-first-kind.html General Basic Lagrangian distribution of the first kind (GLD1)]: $P(X=0)=f(0) , * [http://socr.ucla.edu/htmls/dist/General-Basic-Lagrangian-distribution-of-the-first-kind.html General Basic Lagrangian distribution of the first kind (GLD1)]: [itex] P(X=0)=f(0) , Line 252: Line 252: * [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:[itex] n=2 \$ * [http://socr.ucla.edu/htmls/dist/Triangular_Distribution.html TSP to triangular]:$n=2 \$ * [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:$\kappa \to 0 \$ * [http://socr.ucla.edu/htmls/dist/Uniform_Distribution.html von Mises to Uniform]:$\kappa \to 0 \$ + * [http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html Lévy to Cauchy]:$\alpha =1 \$ + * [http://socr.ucla.edu/htmls/dist/Gaussian_Distribution.html Lévy to Gaussian]:$\alpha \to 2$ + * [http://socr.ucla.edu/htmls/dist/Power-series_Distribution.html Modified Power Series to Power series]:$u(c)=c \$ + * [http://socr.ucla.edu/htmls/dist/Geometric_Distribution.html BLD1 to Geometric]:$g(z)=1-p+pz \$ where[itex]0

## 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
• 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
• 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
• 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
• Rayleigh: $f(x) = \frac{2x}{\alpha}e^{-x^2/\alpha}. x>0 \!$
• Standard Triangular: $f(x) = \begin{cases} x+1, -1
• 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
• 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
• Extreme value: $f(x)=(\beta/\alpha)e^{x\beta-e^{x\beta}/\alpha}. -\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
• 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
• Kolmogorov-Smirnov: $\!$
• Exponential Power: $f(x)=(e^{1-e^{\lambda x^\kappa}})e^{\lambda x^\kappa}\lambda \kappa x^{\kappa-1}. x>0 \!$
• 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: $P(X=x)={n\over x}\binom{{mx}}{x-n}p^{x-n} q^{n+mx-x}$ 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 + θx)x − 1e − (M + θx) / 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) = qn , $P(X=x)=npq^{n-1} {\Gamma (kx+x-1)\over x!\Gamma (kx)} \left({P\over Q} \right)^{x-1} Q^{-kx} {}_{2} F_{1} [1-x,1-n;2-x-kx;{-pQ\over Pq} ]$ 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) = (eθ − 1) − 1θx / 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
• Misrecorded Poisson distribution: $P(X=0)=\omega +(1-\omega )e^{-\theta }, P(X=x)=(1-\omega ){e^{-\theta } \theta ^{x} \over x!} , x\ge 1$