\renewcommand{\coddate}{{\small \today}}
\renewcommand{\sheweek}{Summary of distributions}


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\section*{Discrete Distributions}
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\begin{enumerate}
\item \textbf{Degenerate}: all probability concentrated at a
single point.
\begin{itemize}
\item $f(x) =1 \quad$ for $x=a$.
\item $M(t) = e^{at}, \quad K(t) = at$.
\item $\mu = a, \quad \sigma^2 = 0$.
\end{itemize}

\item \textbf{Bernoulli}: trials with two possible
outcomes, here labeled $X=0$ and $X=1$. Equivalent to a binomial
distribution with $n=1$.
\begin{itemize}
\item $f(x) = p^x(1-p)^{1-x} \quad $ for $x=0, \ 1$.
\item $M(t) = 1 - p + p e^t, \quad K(t) = \log (1-p+pe^t)$.
\item $\mu = p, \quad \sigma^2 = p(1-p)$.
\end{itemize}

\item \textbf{Binomial}: number of successes
in $n$ independent Bernoulli trials with probability of success
$p$. Notation $\bin(n,p)$.
\begin{itemize}
\item $f(x) =  \frac{n!}{x!(n-x)!} p^x(1-p)^{n-x} \quad$ for $x=0, \ 1, \ldots, \ n$.
\item $M(t) = (1 - p + p e^t)^n, \quad K(t) = n \log (1-p+pe^t)$.
\item $\mu = n p, \quad \sigma^2 = n p(1-p)$.
\end{itemize}

\item \textbf{Geometric}: number of trials before first success in
repeated independent Bernoullis. Equivalent to a negative binomial
with $r=1$.
\begin{itemize}
\item $f(x) =  (1-p)^{x-1} p \quad$ for $x=1, \ 2, \ldots$.
\item $M(t) = \frac{p e^t}{1 - (1- p) e^t},
\quad K(t) = - \log \{(1-\frac{1}{p}) + \frac{1}{p} e^{-t}\}
\quad$ for $|t| < - \log(1-p)$.
\item $\mu = \frac{1}{p}, \quad \sigma^2 = \frac{1-p}{p^2}$.
\end{itemize}

\item \textbf{Negative Binomial}: number of trials before $r^{\rm
th}$ success in repeated independent Bernoullis.
\begin{itemize}
\item $f(x) =  \binom{x-1}{r-1} p^r (1-p)^{x-r}\quad$ for $x=r, \ r+1, \ldots$.
\item $M(t) = \left(\frac{p e^t}{1 - (1- p) e^t}\right)^r,
\quad K(t) = - r \log \{(1-\frac{1}{p}) + \frac{1}{p} e^{-t}\}
\quad$ for $|t| < - \log(1-p)$.
\item $\mu = \frac{r}{p}, \quad \sigma^2 = \frac{r(1-p)}{p^2}$.
\end{itemize}

\item \textbf{Poisson}: often used for the number of events which
occur in an interval of time. Notation $\pois(\lambda)$.
\begin{itemize}
\item $f(x) = \frac{\lambda^x e^{-\lambda}}{x!} \quad$ for $x=0, \ 1, \ldots$.
\item $M(t) = e^{\lambda(e^t-1)},
\quad K(t) = \lambda(e^t-1)$.
\item $\mu = \lambda, \quad \sigma^2 = \lambda$.
\end{itemize}

\item \textbf{Hypergeometric}: number of type 1 chosen when
selected $n$ without replacement from urn containing a total of
$N$ made up of $N_1$ of type 1 and $N_2$ of type 2.
\begin{itemize}
\item $f(x) =  \binom{N_1}{x} \binom{N_2}{n-x} / \binom {N}{n} \quad$
for $x \in \{0,\ldots,n\} \cap \{n-N_2,\ldots,N_1\}$.
\item $\mu = n \frac{N_1}{N}, \quad
\sigma^2 = n \frac{N_1}{N} \frac{N_2}{N} \frac{N-n}{N-1}$.
\end{itemize}

\item \textbf{Uniform}: choosing from $m$ with equal probability.
\begin{itemize}
\item $f(x) = \frac{1}{m} \quad$ for $x=1, \ 2, \ldots, m$.
\item $\mu = \frac{m+1}{2}, \quad \sigma^2 = \frac{m^2-1}{12}$.
\end{itemize}
\end{enumerate}

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\section*{Continuous Distributions}
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\begin{enumerate}
\item \textbf{Uniform}: random number chosen from a given closed
interval $[a, \ b]$. Notation $U(a, b)$.
\begin{itemize}
\item $f(x) = \frac{1}{b-a} \quad$ for $a \leq x \leq b$.
\item $M(t) = \frac{e^{tb} - e^{ta}}{t(b-a)} \ $ for $t \neq 0
\quad$ and $M(0)=1$.
\item $\mu = \frac{a+b}{2}, \quad \sigma^2 = \frac{(b-a)^2}{12}$.
\end{itemize}

\item \textbf{Normal}: characterized by first two moments.
Notation $N(\mu,\sigma^2)$. \newline If $X \sim N(\mu,\sigma^2)$
then $(X-\mu)/\sigma = Z \sim N(0,1)$, a standard normal
distribution. \newline A sequence of $r$ independent normals is
written $X_j \sim iN(\mu,\sigma^2)$ for $j = 1,\ldots,r$.
\begin{itemize}
\item $f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^2/(2 \sigma^2)} \quad$ for $-\infty < x < \infty$.
\item $M(t) = e^{\mu t + \sigma^2 t^2/2}$.
\item $E(X) = \mu, \quad \var(X) = \sigma^2$.
\end{itemize}

\item \textbf{Exponential}: waiting time between events when
observing a Poisson process. Notation $\Exp(\theta)$ (somewhat
ambiguous). Equivalent to a gamma distribution with $\alpha = 1$.
\begin{itemize}
\item $f(x) = \theta e^{-\theta x} \quad$ for $0 \leq x < \infty$.
\item $M(t) = \frac{\theta}{\theta-t} \ $ for $t < \theta$.
\item $\mu = 1/\theta, \quad \sigma^2 = 1/\theta^2$.
\end{itemize}

\item \textbf{Chi-square}: If $Z_j \sim iN(0,1)$ then $X =
\sum_{j=1}^r Z_j$ has a chi-square distribution on $r$ degrees of
freedom. Notation $\chi_r^2$ or $\chi^2(r)$. Equivalent to a gamma
distribution with $\alpha =r/2$ and $\theta =1/2$.
\begin{itemize}
\item $f(x) = \frac{1}{\Gamma(r/2) 2^{r/2}} x^{r/2-1} e^{-x/2} \quad$ for $0 \leq x < \infty$.
\item $M(t) = \frac{1}{(1-2 t)^{r/2}} \quad $ for $t < 1/2$.
\item $\mu = r, \quad \sigma^2 = 2r$.
\end{itemize}

\item \textbf{Gamma}: characterized by parameters $\alpha>0$ and
$\theta$. The gamma function is defined by
$$
\Gamma(t) = \int_0^\infty y^{t-1} e^{-y} dy, \quad {\rm for} \
t>0.
$$
Properties of the gamma function $\Gamma(t) = (t-1) \Gamma(t-1)$
for $t>1$ and $\Gamma(n) = (n-1)!$ for positive integer $n$.
Notation for the gamma distribution; $\gam(\alpha,\theta)$ or
$G(\alpha,\theta)$.
\begin{itemize}
\item $f(x) = \frac{1}{\Gamma(\alpha)} \theta^{\alpha} x^{\alpha-1} e^{-\theta x} \quad$ for $0 \leq x < \infty$.
\item $M(t) = \frac{1}{(1-t/\theta)^{\alpha}} \quad $ for $t < \theta$.
\item $\mu = \alpha/\theta, \quad \sigma^2 = \alpha/\theta^2$.
\end{itemize}

\item \textbf{Beta}: characterized by parameters $p>0$ and $q>0$.
The beta function is defined by
$$
B(p,q) = \int_0^1 y^{p-1} (1-y)^{q-1} dy = \frac{\Gamma(p)
\Gamma(q)}{\Gamma(p+q)}.
$$
Notation $\bet(p,q)$.
\begin{itemize}
\item $f(x) = \frac{1}{B(p,q)} x^{p-1} (1-x)^{q-1} \quad$ for $0 \leq x \leq 1$.
\item $\mu = \frac{p}{p+q}, \quad \sigma^2 = \frac{pq}{(p+q+1)(p+q)^2}$.
\end{itemize}

\end{enumerate}
