Appendix A — Distribution Tables
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This appendix provides comprehensive tables of all the named distributions discussed in this book. Each distribution entry also includes a reference to the proposition, example, or exercise where the relevant formula was derived.
A.1 Discrete Distributions
| Distribution | PMF \(f(x)\) | Expected Value | Variance | MGF \(M(t)\) |
|---|---|---|---|---|
| \(\text{Bernoulli}(p)\) | \(\displaystyle\begin{cases} p & x = 1 \\ 1-p & x = 0 \end{cases}\) Definition 8.5 |
\(\displaystyle p\) Example 14.3 |
\(\displaystyle p(1-p)\) Example 15.5 |
\(\displaystyle 1-p+pe^t\) Exercise 34.1 |
| \(\text{Binomial}(n,p)\) | \(\displaystyle\binom{n}{x}p^x(1-p)^{n-x};\) \(x = 0,1,\ldots,n\) Definition 8.6 |
\(\displaystyle np\) Proposition 9.2, Example 14.3 |
\(\displaystyle np(1-p)\) Proposition 11.3, Example 15.5 |
\(\displaystyle (1-p+pe^t)^n\) Exercise 34.1 |
| \(\text{Geometric}(p)\) | \(\displaystyle p(1-p)^{x-1};\) \(x = 1,2,\ldots\) Definition 8.7 |
\(\displaystyle\frac{1}{p}\) Example 9.7 |
\(\displaystyle\frac{1-p}{p^2}\) Proposition 12.2 |
\(\displaystyle\frac{pe^t}{1-(1-p)e^t};\) \(t < -\ln(1-p)\) Exercise 34.4 |
| \(\text{NegativeBinomial}\) \((r,p)\) |
\(\displaystyle\binom{x-1}{r-1}p^r(1-p)^{x-r};\) \(x = r,r+1,\ldots\) Definition 12.2 |
\(\displaystyle\frac{r}{p}\) Proposition 12.2 |
\(\displaystyle\frac{r(1-p)}{p^2}\) Proposition 12.2 |
\(\displaystyle\left(\frac{pe^t}{1-(1-p)e^t}\right)^r;\) \(t < -\ln(1-p)\) Exercise 34.4 |
| \(\text{Poisson}(\mu)\) | \(\displaystyle\frac{e^{-\mu}\mu^x}{x!};\,x = 0,1,2,\ldots\) Definition 12.3 |
\(\displaystyle\mu\) Proposition 12.3 |
\(\displaystyle\mu\) Proposition 12.3 |
\(\displaystyle e^{\mu(e^t-1)}\) Example 34.1 |
| \(\text{Hypergeometric}\) \((n,M,N)\) |
\(\displaystyle\frac{\binom{M}{x}\binom{N-M}{n-x}}{\binom{N}{n}};\) \(x = 0, 1, \dots, n\) Definition 12.1 |
\(\displaystyle\frac{nM}{N}\) Proposition 12.1, Example 14.4 |
\(\frac{nM}{N}\left(1-\frac{M}{N}\right)\frac{N-n}{N-1}\) Proposition 12.1, Example 15.6 |
no simple expression |
A.2 Continuous Distributions
| Distribution | PDF \(f(x)\) | Expected Value | Variance | MGF \(M(t)\) |
|---|---|---|---|---|
| \(\text{Uniform}(a,b)\) | \(\displaystyle\frac{1}{b-a};\, a < x < b\) Definition 22.1 |
\(\displaystyle\frac{a+b}{2}\) Proposition 22.2 |
\(\displaystyle\frac{(b-a)^2}{12}\) Proposition 22.2 |
\(\displaystyle\frac{e^{bt}-e^{at}}{(b-a)t}\) Exercise 34.2 |
| \(\text{Exponential}(\lambda)\) | \(\displaystyle\lambda e^{-\lambda x};\, x > 0\) Definition 22.2 |
\(\displaystyle\frac{1}{\lambda}\) Proposition 22.4 |
\(\displaystyle\frac{1}{\lambda^2}\) Proposition 22.4 |
\(\displaystyle\frac{\lambda}{\lambda-t}; t < \lambda\) Example 34.3 |
| \(\text{Normal}(\mu,\sigma^2)\) | \(\displaystyle\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}};\) \(-\infty < x < \infty\) Definition 22.4 |
\(\displaystyle\mu\) Proposition 22.6 |
\(\displaystyle\sigma^2\) Proposition 22.6 |
\(\displaystyle e^{\mu t+\frac{\sigma^2t^2}{2}}\) Example 34.2 |
| \(\text{Gamma}(\alpha,\lambda)\) | \(\displaystyle\frac{\lambda^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\lambda x};\, x > 0\) Definition 39.2 |
\(\displaystyle\frac{\alpha}{\lambda}\) Proposition 39.2 |
\(\displaystyle\frac{\alpha}{\lambda^2}\) Proposition 39.2 |
\(\displaystyle\left(\frac{\lambda}{\lambda-t}\right)^\alpha;\, t < \lambda\) Proposition 39.3 |
| \(\text{Chi-square}(k)\) | \(\displaystyle\frac{1}{2^{k/2}\Gamma(k/2)}x^{k/2-1}e^{-x/2};\) \(x > 0\) Note A.1 |
\(\displaystyle k\) Note A.1 |
\(\displaystyle 2k\) Note A.1 |
\(\displaystyle\left(\frac{1}{1-2t}\right)^{k/2};\, t < \frac{1}{2}\) Note A.1 |
| \(\text{Beta}(\alpha,\beta)\) | \(\displaystyle\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1};\) \(0 < x < 1\) Definition 42.1 |
\(\displaystyle\frac{\alpha}{\alpha+\beta}\) Proposition 42.2 |
\(\displaystyle\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\) Proposition 42.2 |
no simple expression |