\renewcommand{\coddate}{{\small 6.10.2003}}
\renewcommand{\sheweek}{Discrete 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}
