s23-probability-theory/inputs/a_0_distributions.tex

\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}
}