1 Discrete Distributions

Overview. This section summarizes selected discrete probability distributions used throughout Loss Data Analytics. Relevant functions and R code are provided.

1.1 The (a,b,0) Class

Poisson

Functions

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Name} & \text{Function} \\ \hline ~~p_0 & e^{-\lambda} \\ \hline \small{\text{Probability mass function}} & \frac{e^{-\lambda}\lambda^k}{k!} \\ ~~p_k & \\ \hline \small{\text{Expected value}} & \lambda \\ ~~\mathrm{E}[N] & \\ \hline \small{\text{Variance}} & \lambda \\ \hline \small{\text{Probability generating function}} & e^{\lambda(z-1)} \\ ~~P(z) & \\ \hline a \small{\text{ and }} b \small{\text{ for recursion}} & a=0 \\ & b=\lambda \\ \hline \end{array} \end{matrix} \]

R Commands

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Function Name} & \text{R Command} \\ \hline \small{\text{Probability mass function}} & \text{dpois}(x=, lambda=\lambda) \\ \hline \small{\text{Distribution function}} & \text{ppois}(p=, lambda=\lambda) \\ \hline \small{\text{Quantile function}} & \text{qpois}(q=, lambda=\lambda) \\ \hline \small{\text{Random sampling function}} & \text{rpois}(n=, lambda=\lambda) \\ \hline \end{array} \end{matrix} \]

Geometric

Functions

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Name} & \text{Function} \\ \hline ~~p_0 & \frac{1}{1+\beta} \\ \hline \small{\text{Probability mass function}} & \frac{\beta^k}{(1+\beta)^{k+1}} \\ ~~p_k & \\ \hline \small{\text{Expected value}} & \beta \\ ~~\mathrm{E}[N] & \\ \hline \small{\text{Variance}} & \beta(1+\beta) \\ \hline \small{\text{Probability generating function}} & [1-\beta(z-1)]^{-1} \\ ~~P(z) & \\ \hline a \small{\text{ and }} b \small{\text{ for recursion}} & a=\frac{\beta}{1+\beta} \\ & b=0 \\ \hline \end{array} \end{matrix} \]

R Commands

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Function Name} & \text{R Command} \\ \hline \small{\text{Probability mass function}} & \text{dgeom}(x=, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Distribution function}} & \text{pgeom}(p=, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Quantile function}} & \text{qgeom}(q=, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Random sampling function}} & \text{rgeom}(n=, prob=\frac{1}{1+\beta}) \\ \hline \end{array} \end{matrix} \]

Binomial

Functions

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Name} & \text{Function} \\ \hline \small{\text{Parameter assumptions}} & 0<q<1,~\text{m is an integer} \\ & 0 \leq k \leq m\\ \hline ~~p_0 &(1-q)^m \\ \hline \small{\text{Probability mass function}} & \binom{m}{k}q^k(1-q)^{m-k} \\ ~~p_k & \\ \hline \small{\text{Expected value}} & mq \\ ~~\mathrm{E}[N] & \\ \hline \small{\text{Variance}} & mq(1-q) \\ \hline \small{\text{Probability generating function}} & [1+q(z-1)]^m \\ ~~P(z) & \\ \hline a \small{\text{ and }} b \small{\text{ for recursion}} & a=\frac{-q}{1-q} \\ & b=\frac{(m+1)q}{1-q} \\ \hline \end{array} \end{matrix} \]

R Commands

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Function Name} & \text{R Command} \\ \hline \small{\text{Probability mass function}} & \text{dbinom}(x=, size=m, prob=p) \\ \hline \small{\text{Distribution function}} & \text{pbinom}(p=, size=m, prob=p) \\ \hline \small{\text{Quantile function}} & \text{qbinom}(q=, size=m, prob=p) \\ \hline \small{\text{Random sampling function}} & \text{rbinom}(n=, size=m, prob=p) \\ \hline \end{array} \end{matrix} \]

Negative Binomial

Functions

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Name} & \text{Function} \\ \hline ~~p_0 & (1+\beta)^{-r} \\ \hline \small{\text{Probability mass function}} & \frac{r(r+1)\cdots(r+k-1)\beta^k}{k!(1+\beta)^{r+k}} \\ ~~p_k & \\ \hline \small{\text{Expected value}} & r\beta \\ ~~\mathrm{E}[N] & \\ \hline \small{\text{Variance}} & r\beta(1+\beta) \\ \hline \small{\text{Probability generating function}} & [1-\beta(z-1)]^{-r} \\ ~~P(z) & \\ \hline a \small{\text{ and }} b \small{\text{ for recursion}} & a=\frac{\beta}{1+\beta} \\ & b=\frac{(r-1)\beta}{1+\beta} \\ \hline \end{array} \end{matrix} \]

R Commands

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Function Name} & \text{R Command} \\ \hline \small{\text{Probability mass function}} & \text{dnbinom}(x=, size=r, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Distribution function}} & \text{pnbinom}(p=, size=r, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Quantile function}} & \text{qnbinom}(q=, size=r, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Random sampling function}} & \text{rnbinom}(n=, size=r, prob=\frac{1}{1+\beta}) \\ \hline \end{array} \end{matrix} \]

1.2 The (a,b,1) Class

Zero Truncated Poisson

Functions

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Name} & \text{Function} \\ \hline ~~p^T_1 & \frac{\lambda}{e^\lambda-1} \\ \hline \small{\text{Probability mass function}} & \frac{\lambda^k}{k!(e^\lambda-1)} \\ ~~p^T_k & \\ \hline \small{\text{Expected value}} & \frac{\lambda}{1-e^{-\lambda}} \\ ~~\mathrm{E}[N] & \\ \hline \small{\text{Variance}} & \frac{\lambda[1-(\lambda+1)e^{-\lambda}]}{(1-e^{-\lambda})^2} \\ \hline \tilde\lambda & \ln(\frac{n\hat\mu}{n_1}) \\ \hline \small{\text{Probability generating function}} & \frac{e^{\lambda z}-1}{e^\lambda-1} \\ ~~P(z) & \\ \hline a \small{\text{ and }} b \small{\text{ for recursion}} & a=0 \\ & b=\lambda \\ \hline \end{array} \end{matrix} \]

R Commands

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Function Name} & \text{R Command} \\ \hline \small{\text{Probability mass function}} & \text{dztpois}(x=, lambda=\lambda) \\ \hline \small{\text{Distribution function}} & \text{pztpois}(p=, lambda=\lambda) \\ \hline \small{\text{Quantile function}} & \text{qztpois}(q=, lambda=\lambda) \\ \hline \small{\text{Random sampling function}} & \text{rztpois}(n=, lambda=\lambda) \\ \hline \end{array} \end{matrix} \]

Zero Truncated Geometric

Functions

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Name} & \text{Function} \\ \hline ~~p^T_1 & \frac{1}{1+\beta} \\ \hline \small{\text{Probability mass function}} & \frac{\beta^{k-1}}{(1+\beta)^k} \\ ~~p^T_k & \\ \hline \small{\text{Expected value}} & 1+\beta \\ ~~\mathrm{E}[N] & \\ \hline \small{\text{Variance}} & \beta(1+\beta) \\ \hline \tilde\beta & \hat\mu-1 \\ \hline \small{\text{Probability generating function}} & \frac{[1-\beta(z-1)]^{-1}-(1+\beta)^{-1}}{1-(1+\beta)^{-1}} \\ ~~P(z) & \\ \hline a \small{\text{ and }} b \small{\text{ for recursion}} & a=\frac{\beta}{1+\beta} \\ & b=0 \\ \hline \end{array} \end{matrix} \]

R Commands

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Function Name} & \text{R Command} \\ \hline \small{\text{Probability mass function}} & \text{dztgeom}(x=, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Distribution function}} & \text{pztgeom}(p=, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Quantile function}} & \text{qztgeom}(q=, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Random sampling function}} & \text{rztgeom}(n=, prob=\frac{1}{1+\beta}) \\ \hline \end{array} \end{matrix} \]

Zero Truncated Binomial

Functions

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Name} & \text{Function} \\ \hline \small{\text{Parameter assumptions}} & 0<q<1,~\text{m is an integer} \\ & 0 \leq k \leq m\\ \hline ~~p^T_1 & \frac{m(1-q)^{m-1}q}{1-(1-q)^m} \\ \hline \small{\text{Probability mass function}} & \frac{\binom{m}{k}q^k(1-q)^{m-k}}{1-(1-q)^m} \\ ~~p^T_k & \\ \hline \small{\text{Expected value}} & \frac{mq}{1-(1-q)^m} \\ ~~\mathrm{E}[N] & \\ \hline \small{\text{Variance}} & \frac{mq[(1-q)-(1-q+mq)(1-q)^m]}{[1-(1-q)^m]^2} \\ \hline \tilde q & \frac{\hat\mu}{m} \\ \hline \small{\text{Probability generating function}} & \frac{[1+q(z-1)^m]-(1-q)^m}{1-(1-q)^m} \\ ~~P(z) & \\ \hline a \small{\text{ and }} b \small{\text{ for recursion}} & a=\frac{1}{1-q} \\ & b=\frac{(m+1)q}{1-q} \\ \hline \end{array} \end{matrix} \]

R Commmands

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Function Name} & \text{R Command} \\ \hline \small{\text{Probability mass function}} & \text{dztbinom}(x=, size=m, prob=p) \\ \hline \small{\text{Distribution function}} & \text{pztbinom}(p=, size=m, prob=p) \\ \hline \small{\text{Quantile function}} & \text{qztbinom}(q=, size=m, prob=p) \\ \hline \small{\text{Random sampling function}} & \text{rztbinom}(n=, size=m, prob=p) \\ \hline \end{array} \end{matrix} \]

Zero Truncated Negative Binomial

Functions

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Name} & \text{Function} \\ \hline \small{\text{Parameter assumptions}} & r>-1, r\neq0 \\ \hline ~~p^T_1 & \frac{r\beta}{(1+\beta)^{r+1}-(1+\beta)} \\ \hline \small{\text{Probability mass function}} & \frac{r(r+1)\cdots(r+k-1)}{k![(1+\beta)^r-1]}(\frac{\beta}{1+\beta})^k \\ ~~p^T_k & \\ \hline \small{\text{Expected value}} & \frac{r\beta}{1-(1+\beta)^{-r}} \\ ~~\mathrm{E}[N] & \\ \hline \small{\text{Variance}} & \frac{r\beta[(1+\beta)-(1+\beta+r\beta)(1+\beta)^{-r}]}{[1-(1+\beta)^{-r}]^2} \\ \hline \tilde\beta & \frac{\hat\sigma^2}{\hat\mu}-1 \\ \hline \tilde r & \frac{\hat\mu^2}{\hat\sigma^2-\hat\mu} \\ \hline \small{\text{Probability generating function}} & \frac{[1-\beta(z-1)]^{-r}-(1+\beta)^{-r}}{1-(1+\beta)^{-r}} \\ ~~P(z) & \\ \hline a \small{\text{ and }} b \small{\text{ for recursion}} & a=\frac{\beta}{1+\beta} \\ & b=\frac{(r-1)\beta}{1+\beta} \\ \hline \end{array} \end{matrix} \]

R Commands

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Function Name} & \text{R Command} \\ \hline \small{\text{Probability mass function}} & \text{dztnbinom}(x=, size=r, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Distribution function}} & \text{pztnbinom}(p=, size=r, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Quantile function}} & \text{qztnbinom}(q=, size=r, prob=\frac{1}{1+\beta}) \\ \hline \small{\text{Random sampling function}} & \text{rztnbinom}(n=, size=r, prob=\frac{1}{1+\beta}) \\ \hline \end{array} \end{matrix} \]

Logarithmic

Functions

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Name} & \text{Function} \\ \hline ~~p^T_1 & \frac{\beta}{(1+\beta)ln(1+\beta)} \\ \hline \small{\text{Probability mass function}} & \frac{\beta^k}{k(1+\beta)^k \ln (1+\beta)} \\ ~~p^T_k & \\ \hline \small{\text{Expected value}} & \frac{\beta}{\ln (1+\beta)} \\ ~~\mathrm{E}[N] & \\ \hline \small{\text{Variance}} & \frac{\beta[1+\beta-\frac{\beta}{ln(1+\beta)}]}{\ln (1+\beta)} \\ \hline \tilde\beta & \frac{n\hat\mu}{n_1}-1~=~ \frac{2(\hat\mu-1)}{\hat\mu} \\ \hline \small{\text{Probability generating function}} & 1-\frac{ln[1-\beta(z-1)]}{\ln (1+\beta)} \\ ~~P(z) & \\ \hline a \small{\text{ and }} b \small{\text{ for recursion}} & a=\frac{\beta}{1+\beta} \\ & b=\frac{-\beta}{1+\beta} \\ \hline \end{array} \end{matrix} \]

R Commands

\[ \begin{matrix} \begin{array}{l|c} \hline \text{Function Name} & \text{R Command} \\ \hline \small{\text{Probability mass function}} & \text{dnbinom}(x=,prob=\frac{\beta}{1+\beta}) \\ \hline \small{\text{Distribution function}} & \text{pnbinom}(p=,prob=\frac{\beta}{1+\beta}) \\ \hline \small{\text{Quantile function}} & \text{qnbinom}(q=,prob=\frac{\beta}{1+\beta}) \\ \hline \small{\text{Random sampling function}} & \text{rnbinom}(n=,prob=\frac{\beta}{1+\beta}) \\ \hline \end{array} \end{matrix} \]