82 lines
3.0 KiB
TeX
82 lines
3.0 KiB
TeX
\subsection{List of Distributions}
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{\rowcolors{2}{gray!10}{white}
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\begin{longtable}{llllllll}
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& Symbol & Mass (PMF) & Distribution (CDF) & $\bE$ & $\Var$ & $\phi_X(t) = \bE[e^{\i t X}]$ & $M_X(t) = \bE[e^{tX}]$ \\
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\hline
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Deterministic &
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$\delta_a$ &
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$\One_{x = a}$&
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$\One_{[a,\infty)}$ &
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$a$ &
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$0$ &
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$e^{\i t a}$ &
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$e^{t a}$ \\
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Bernoulli &
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$\Bin(1,p)$ & & & & & &\\
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Binomial &
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$\Bin(n,p)$ &
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$\binom{n}{k} p^{k} (1-p)^{n-k}$ &
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$\sum_{j=0}^{\left\lfloor x \right\rfloor} \binom{n}{j} p^{j} (1-p)^{n-j}$ &
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$n p$&
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$n p (1-p)$ &
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$((1-p) + pe^{\i t})^n$ &
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$((1-p) + pe^t)^n$\\
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Geometric &
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$\Geo(p)$ &
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$(1-p)^{k-1} p$ &
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$1 - (1 - p)^{\left\lfloor x \right\rfloor}$ &
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$\frac{1}{p}$ &
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$\frac{1-p}{p^2}$&
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$\frac{p e^{\i t}}{1 - (1 -p)e^{\i t}}$ &
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$\frac{p e^{t}}{1 - (1 -p)e^{t}}$\\
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Poisson &
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$\Poi(\lambda)$ &
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$\frac{\lambda^k e^{-\lambda}}{k!}$ &
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$e^{-\lambda} \sum_{j=0}^{\left\lfloor x \right\rfloor} \frac{\lambda^j}{j!}$ &
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$\lambda$ &
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$\lambda$ &
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$e^{\lambda (e^{\i t} -1)}$ &
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$e^{\lambda (e^{t} -1)}$\\
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\end{longtable}
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}
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{\rowcolors{2}{gray!10}{white}
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\begin{longtable}{llllllll}
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& Symbol & Density (PDF) & Distribution (CDF) & $\bE$ & $\Var$ & $\phi_X(t) = \bE[e^{\i t X}]$ & $M_X(t) = \bE[e^{tX}]$ \\
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\hline
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Uniform &
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$\Unif([a,b])$ &
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$\frac{1}{b-a} \One_{[a,b]}$ &
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$\frac{x-a}{b-a} \One_{[a,b]} + \One_{(b,\infty)}$ &
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$\frac{a+b}{2}$ &
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$\frac{(b-a)^2}{12}$ &
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$\frac{e^{\i t b} - e^{\i t a}}{\i t (b-a)}$\footnote{$\phi_X(0) = 1$ }&
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$\frac{e^{t b} - e^{t a}}{t (b-a)}$\footnote{$M_X(0) = 1$}\\
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Exponential &
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$\Exp(\lambda)$ &
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$\One_{x \ge 0}\lambda e^{-\lambda x}$ &
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$\One_{x \ge 0} (1 - e^{-\lambda x})$ &
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$\frac{1}{\lambda}$ &
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$\frac{1}{\lambda^2}$ &
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$\frac{\lambda}{\lambda - \i t}$ &
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$\frac{\lambda}{\lambda - t}, t < \lambda$\\
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Cauchy &
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$\Cauchy(x_0, \gamma)$ &
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$\frac{1}{\pi \gamma \left( 1 + \left( \frac{x - x_0}{\gamma} \right)^2 \right) }$ &
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$\frac{1}{\pi} \arctan \left( \frac{x - x_0}{\gamma} \right) + \frac{1}{2}$ &
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n/a &
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n/a &
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$e^{x_0 \i t - \gamma |t|}$ &
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n/a \\
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Normal &
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$\cN(\mu, \sigma)$ &
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$\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(\mu - x)^2}{2 \sigma^2}}$ &
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$\Phi\left( \frac{x - \mu}{\sigma} \right) $ &
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$\mu$ &
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$\sigma^2$ &
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$e^{\i \mu t - \frac{\sigma^2 t^2}{2}}$ &
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$e^{\mu t + \frac{\sigma^2 t^2}{2}}$\\
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\end{longtable}
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}
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