s23-probability-theory/inputs/a_0_distributions.tex
2023-07-06 00:37:02 +02:00

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