17.2 Appendix c: data on some common distributions

Discrete distributions

1. Indicator functions $X=I_{E}P(X=1)=P\left(E\right)=pP(X=0)=q=1-p$
   $$E\left[X\right]=p\mathrm{Var}\left[X\right]=pq{M}_X\left(s\right)=q+p{e}^s{g}_X\left(s\right)=q+ps$$
2. Simple random variable ${\displaystyle X=\sum _{i=1}^n{t}_i{I}_{A_i}\text{(a}\text{primitive}\text{form)}P\left(A_i\right)=p_i}$
   $$E\left[X\right]=\sum _{i=1}^n{t}_i{p}_i\mathrm{Var}\left[X\right]=\sum _{i=1}^n{t}_i^2{p}_i{q}_i-2\sum _{i<j}{t}_i{t}_j{p}_i{p}_j{M}_X\left(s\right)=\sum _{i=1}^n{p}_i{e}^{s{t}_i}$$
3. Binomial $(n,p)$ ${\displaystyle X=\sum _{i=1}^n{I}_{E_i}\text{with}\{I_{E_i}:1\le i\le n\}\text{iid}P\left(E_i\right)=p}$
   $$P(X=k)=C(n,k)p^k{q}^{n-k}$$
   $$E\left[X\right]=np\mathrm{Var}\left[X\right]=npq{M}_X\left(s\right)={(q+p{e}^s)}^n{g}_X\left(s\right)={(q+ps)}^n$$
   MATLAB :          $P(X=k)=\text{ibinom}(n,p,k)P(X\ge k)=\text{cbinom}(n,p,k)$
4. Geometric $\left(p\right)$ $P(X=k)=p{q}^k\forall k\ge 0$
   $$E\left[X\right]=q/p\mathrm{Var}\left[X\right]=q/p^2{M}_X\left(s\right)=\frac{p}{1-q{e}^s}{g}_X\left(s\right)=\frac{p}{1-qs}$$
   If $Y-1\sim$ geometric $\left(p\right)$ , so that $P(Y=k)=p{q}^{k-1}\forall k\ge 1$ , then
   $$E\left[Y\right]=1/p\mathrm{Var}\left[X\right]=q/p^2{M}_Y\left(s\right)=\frac{p{e}^s}{1-q{e}^s}{g}_Y\left(s\right)=\frac{ps}{1-qs}$$
5. Negative binomial $(m,p)$ . X is the number of failures before the m th success. $P(X=k)=C(m+k-1,m-1)p^m{q}^k\forall k\ge 0$ .
   $$E\left[X\right]=mq/p\mathrm{Var}\left[X\right]=mq/p^2{M}_X\left(s\right)={\left(\frac{p}{1-q{e}^s}\right)}^m{g}_X\left(s\right)={\left(\frac{p}{1-qs}\right)}^m$$
   For $Y_m=X_m+m$ , the number of the trial on which m th success occurs. $P(Y=k)=C(k-1,m-1)p^m{q}^{k-m}\forall k\ge m$ .
   $$E\left[Y\right]=m/p\mathrm{Var}\left[Y\right]=mq/p^2{M}_Y\left(s\right)={\left(\frac{p{e}^s}{1-q{e}^s}\right)}^m{g}_Y\left(s\right)={\left(\frac{ps}{1-qs}\right)}^m$$
   MATLAB :          $P(Y=k)=\text{nbinom}(m,p,k)$
6. Poisson $\left(\mu \right)$ . ${\displaystyle P(X=k)=e^{-\mu }\frac{{\mu }^k}{k!}\forall k\ge 0}$
   $$E\left[X\right]=\mu \mathrm{Var}\left[X\right]=\mu M_X\left(s\right)=e^{\mu (e^s-1)}{g}_X\left(s\right)=e^{\mu (s-1)}$$
   MATLAB :          $P(X=k)=\text{ipoisson}(m,k)P(X\ge k)=\text{cpoisson}(m,k)$

Absolutely continuous distributions

1. Uniform $(a,b)$ ${\displaystyle f_X\left(t\right)=\frac{1}{b-a}a<t<b}$ (zero elsewhere)
   $$E\left[X\right]=\frac{b+a}{2}\mathrm{Var}\left[X\right]=\frac{{(b-a)}^2}{12}{M}_X\left(s\right)=\frac{e^{sb}-e^{sa}}{s(b-a)}$$
2. Symmetric triangular $(-a,a)$ ${\displaystyle f_X\left(t\right)=\left\{\begin{array}{cc}(a+t)/a^2\hfill & -a\le t<0\hfill \\ (a-t)/a^2\hfill & 0\le t\le a\hfill \end{array}\right)}$
   $$E\left[X\right]=0\mathrm{Var}\left[X\right]=\frac{a^2}{6}{M}_X\left(s\right)=\frac{e^{as}+e^{-as}-2}{a^2{s}^2}=\frac{e^{as}-1}{as}·\frac{1-e^{-as}}{as}$$
3. Exponential $\left(\lambda \right)$ $f_X\left(t\right)=\lambda e^{-\lambda t}t\ge 0$
   $$E\left[X\right]=\frac{1}{\lambda }\mathrm{Var}\left[X\right]=\frac{1}{{\lambda }^2}{M}_X\left(s\right)=\frac{\lambda }{\lambda -s}$$
4. Gamma $(\alpha ,\lambda )$ ${\displaystyle f_X\left(t\right)=\frac{{\lambda }^{\alpha }{t}^{\alpha -1}{e}^{-\lambda t}}{\Gamma \left(\alpha \right)}t\ge 0}$
   $$E\left[X\right]=\frac{\alpha }{\lambda }\mathrm{Var}\left[X\right]=\frac{\alpha }{{\lambda }^2}{M}_X\left(s\right)={\left(\frac{\lambda }{\lambda -s}\right)}^{\alpha }$$
   MATLAB :          $P(X\le t)=\text{gammadbn}(\alpha ,\lambda ,t)$
5. Normal $N(\mu ,{\sigma }^2)$ ${\displaystyle f_X\left(t\right)=\frac{1}{\sigma \sqrt{2\pi }}exp\left(-,\frac{1}{2},{\left(\frac{t-\mu }{\sigma }\right)}^2\right)}$
   $$E\left[X\right]=\mu \mathrm{Var}\left[X\right]{\sigma }^2{M}_X\left(s\right)=exp\left(\frac{{\sigma }^2{s}^2}{2},+,\mu ,s\right)$$
   MATLAB :          $P(X\le t)=\text{gaussian}(\mu ,{\sigma }^2,t)$
6. Beta $(r,s)$
   $$f_X\left(t\right)=\frac{\Gamma (r+s)}{\Gamma \left(r\right)\Gamma \left(s\right)}{t}^{r-1}{(1-t)}^{s-1}0<t<1,r>0,s>0$$
   $$E\left[X\right]=\frac{r}{r+s}\text{Var}\left[X\right]=\frac{rs}{{(r+s)}^2(r+s+1)}$$
   MATLAB : $f_X\left(t\right)=\text{beta}(r,s,t)P(X\le t)=\text{betadbn}(r,s,t)$
7. Weibull $(\alpha ,\lambda ,\nu )$
   $$F_X\left(t\right)=1-e^{-\lambda {(t-\nu )}^{\alpha }},\alpha >0,\lambda >0,\nu \ge 0,t\ge \nu$$
   $$E\left[X\right]=\frac{1}{{\lambda }^{1/\alpha }}\Gamma (1+1/\alpha )+\nu \text{Var}\left[X\right]=\frac{1}{{\lambda }^{2/\alpha }}\left[\Gamma ,(1+2/\lambda ),-,{\Gamma }^2,(1+1/\lambda )\right]$$
   MATLAB : $(\nu =0$ only)
   $$f_X\left(t\right)=\text{weibull}(a,l,t)P(X\le t)=\text{weibulld}(a,l,t)$$

Relationship between gamma and poisson distributions

• If $X\sim$ gamma $(n,\lambda )$ , then $P(X\le t)=P(Y\ge n)$ where $Y\sim$ Poisson $\left(\lambda t\right)$ .
• If $Y\sim$ Poisson $\left(\lambda t\right)$ , then $P(Y\ge n)=P(X\le t)$ where $X\sim$ gamma $(n,\lambda )$ .