# 17.2 Appendix c: data on some common distributions

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Data are provided on some commonly used discrete and absolutely continuous distributions. Matlab procedures are provided for some.

## Discrete distributions

1. Indicator functions $X={I}_{E}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(X=1\right)=P\left(E\right)=p\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(X=0\right)=q=1-p$
$E\left[X\right]=p\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=pq\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{X}\left(s\right)=q+p{e}^{s}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{g}_{X}\left(s\right)=q+ps$
2. Simple random variable $X=\sum _{i=1}^{n}{t}_{i}{I}_{{A}_{i}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{(a}\phantom{\rule{4.pt}{0ex}}\text{primitive}\phantom{\rule{4.pt}{0ex}}\text{form)}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left({A}_{i}\right)={p}_{i}$
$E\left[X\right]=\sum _{i=1}^{n}{t}_{i}{p}_{i}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=\sum _{i=1}^{n}{t}_{i}^{2}{p}_{i}{q}_{i}-2\sum _{i
3. Binomial $\left(n,p\right)$ $X=\sum _{i=1}^{n}{I}_{{E}_{i}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{with}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{{I}_{{E}_{i}}:\phantom{\rule{0.166667em}{0ex}}1\le i\le n\right\}\phantom{\rule{0.277778em}{0ex}}\text{iid}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left({E}_{i}\right)=p$
$P\left(X=k\right)=C\left(n,\phantom{\rule{0.166667em}{0ex}}k\right){p}^{k}{q}^{n-k}$
$E\left[X\right]=np\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=npq\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{X}\left(s\right)={\left(q+p{e}^{s}\right)}^{n}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{g}_{X}\left(s\right)={\left(q+ps\right)}^{n}$
MATLAB :          $P\left(X=k\right)=\text{ibinom}\phantom{\rule{0.166667em}{0ex}}\left(n,p,k\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(X\ge k\right)=\text{cbinom}\phantom{\rule{0.166667em}{0ex}}\left(n,p,k\right)$
4. Geometric $\left(p\right)$ $P\left(X=k\right)=p{q}^{k}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}k\ge 0$
$E\left[X\right]=q/p\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=q/{p}^{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{X}\left(s\right)=\frac{p}{1-q{e}^{s}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{g}_{X}\left(s\right)=\frac{p}{1-qs}$
If $Y-1\sim$ geometric $\left(p\right)$ , so that $P\left(Y=k\right)=p{q}^{k-1}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}k\ge 1$ , then
$E\left[Y\right]=1/p\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=q/{p}^{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{Y}\left(s\right)=\frac{p{e}^{s}}{1-q{e}^{s}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{g}_{Y}\left(s\right)=\frac{ps}{1-qs}$
5. Negative binomial $\left(m,p\right)$ . X is the number of failures before the m th success. $P\left(X=k\right)=C\left(m+k-1,\phantom{\rule{0.166667em}{0ex}}m-1\right){p}^{m}{q}^{k}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}k\ge 0$ .
$E\left[X\right]=mq/p\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=mq/{p}^{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{X}\left(s\right)={\left(\frac{p}{1-q{e}^{s}}\right)}^{m}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{g}_{X}\left(s\right)={\left(\frac{p}{1-qs}\right)}^{m}$
For ${Y}_{m}={X}_{m}+m$ , the number of the trial on which m th success occurs. $P\left(Y=k\right)=C\left(k-1,\phantom{\rule{0.166667em}{0ex}}m-1\right){p}^{m}{q}^{k-m}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}k\ge m$ .
$E\left[Y\right]=m/p\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[Y\right]=mq/{p}^{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{Y}\left(s\right)={\left(\frac{p{e}^{s}}{1-q{e}^{s}}\right)}^{m}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{g}_{Y}\left(s\right)={\left(\frac{ps}{1-qs}\right)}^{m}$
MATLAB :          $P\left(Y=k\right)=\text{nbinom}\phantom{\rule{0.166667em}{0ex}}\left(m,p,k\right)$
6. Poisson $\left(\mu \right)$ . $P\left(X=k\right)={e}^{-\mu }\phantom{\rule{0.166667em}{0ex}}\frac{{\mu }^{k}}{k!}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}k\ge 0$
$E\left[X\right]=\mu \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=\mu \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{X}\left(s\right)={e}^{\mu \left({e}^{s}-1\right)}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{g}_{X}\left(s\right)={e}^{\mu \left(s-1\right)}$
MATLAB :          $P\left(X=k\right)=\text{ipoisson}\left(m,k\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(X\ge k\right)=\text{cpoisson}\left(m,k\right)$

## Absolutely continuous distributions

1. Uniform $\left(a,b\right)$ ${f}_{X}\left(t\right)=\frac{1}{b-a}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a (zero elsewhere)
$E\left[X\right]=\frac{b+a}{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=\frac{{\left(b-a\right)}^{2}}{12}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{X}\left(s\right)=\frac{{e}^{sb}-{e}^{sa}}{s\left(b-a\right)}$
2. Symmetric triangular $\left(-a,a\right)$ ${f}_{X}\left(t\right)=\left\{\begin{array}{cc}\left(a+t\right)/{a}^{2}\hfill & -a\le t<0\hfill \\ \left(a-t\right)/{a}^{2}\hfill & \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0\le t\le a\hfill \end{array}\right)$
$E\left[X\right]=0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=\frac{{a}^{2}}{6}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{X}\left(s\right)=\frac{{e}^{as}+{e}^{-as}-2}{{a}^{2}{s}^{2}}=\frac{{e}^{as}-1}{as}·\frac{1-{e}^{-as}}{as}$
3. Exponential $\left(\lambda \right)$ ${f}_{X}\left(t\right)=\lambda \phantom{\rule{0.166667em}{0ex}}{e}^{-\lambda t}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\ge 0$
$E\left[X\right]=\frac{1}{\lambda }\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=\frac{1}{{\lambda }^{2}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{X}\left(s\right)=\frac{\lambda }{\lambda -s}$
4. Gamma $\left(\alpha ,\lambda \right)$ ${f}_{X}\left(t\right)=\frac{{\lambda }^{\alpha }{t}^{\alpha -1}{e}^{-\lambda t}}{\Gamma \left(\alpha \right)}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\ge 0$
$E\left[X\right]=\frac{\alpha }{\lambda }\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]=\frac{\alpha }{{\lambda }^{2}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{X}\left(s\right)={\left(\frac{\lambda }{\lambda -s}\right)}^{\alpha }$
MATLAB :          $P\left(X\le t\right)=\text{gammadbn}\phantom{\rule{0.166667em}{0ex}}\left(\alpha ,\lambda ,t\right)$
5. Normal $N\left(\mu ,{\sigma }^{2}\right)$ ${f}_{X}\left(t\right)=\frac{1}{\sigma \sqrt{2\pi }}\phantom{\rule{0.166667em}{0ex}}exp\left(-,\frac{1}{2},{\left(\frac{t-\mu }{\sigma }\right)}^{2}\right)$
$E\left[X\right]=\mu \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{Var}\phantom{\rule{0.166667em}{0ex}}\left[X\right]\phantom{\rule{4pt}{0ex}}{\sigma }^{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{M}_{X}\left(s\right)=exp\left(\frac{{\sigma }^{2}{s}^{2}}{2},+,\mu ,s\right)$
MATLAB :          $P\left(X\le t\right)=\text{gaussian}\phantom{\rule{0.166667em}{0ex}}\left(\mu ,{\sigma }^{2},t\right)$
6. Beta $\left(r,\phantom{\rule{0.166667em}{0ex}}s\right)$
${f}_{X}\left(t\right)=\frac{\Gamma \left(r+s\right)}{\Gamma \left(r\right)\Gamma \left(s\right)}{t}^{r-1}{\left(1-t\right)}^{s-1}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}00,\phantom{\rule{0.277778em}{0ex}}s>0$
$E\left[X\right]=\frac{r}{r+s}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{Var}\left[X\right]=\frac{rs}{{\left(r+s\right)}^{2}\left(r+s+1\right)}$
MATLAB : ${f}_{X}\left(t\right)=\text{beta}\left(r,\phantom{\rule{0.166667em}{0ex}}s,\phantom{\rule{0.166667em}{0ex}}t\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(X\le t\right)=\text{betadbn}\left(r,\phantom{\rule{0.166667em}{0ex}}s,\phantom{\rule{0.166667em}{0ex}}t\right)$
7. Weibull $\left(\alpha ,\lambda ,\nu \right)$
${F}_{X}\left(t\right)=1-{e}^{-\lambda {\left(t-\nu \right)}^{\alpha }},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\alpha >0,\phantom{\rule{0.277778em}{0ex}}\lambda >0,\phantom{\rule{0.277778em}{0ex}}\nu \ge 0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\ge \nu$
$E\left[X\right]=\frac{1}{{\lambda }^{1/\alpha }}\phantom{\rule{0.166667em}{0ex}}\Gamma \left(1+1/\alpha \right)+\nu \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{Var}\left[X\right]=\frac{1}{{\lambda }^{2/\alpha }}\phantom{\rule{0.166667em}{0ex}}\left[\Gamma ,\left(1+2/\lambda \right),-,{\Gamma }^{2},\left(1+1/\lambda \right)\right]$
MATLAB : $\left(\nu =0$ only)
${f}_{X}\left(t\right)=\text{weibull}\left(a,\phantom{\rule{0.166667em}{0ex}}l,\phantom{\rule{0.166667em}{0ex}}t\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P\left(X\le t\right)=\text{weibulld}\left(a,\phantom{\rule{0.166667em}{0ex}}l,\phantom{\rule{0.166667em}{0ex}}t\right)$

## Relationship between gamma and poisson distributions

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

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what is the actual application of fullerenes nowadays?
Damian
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is Bucky paper clear?
CYNTHIA
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive