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\title{Probability basics}
\author{\copyright\ Ben Bolker: \today}
\date{}
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\section*{Definitions}

\begin{enumerate*}
\item{In general for two events $A$ and $B$
    the probability of both occurring
    is $\prob(A \cup B) = \prob(A) + \prob(B) - \prob(A \cap B)$,
    where the last bit term is 
    the \emph{joint probability} of $A$ and $B$.
}
\item{For \emph{mutually exclusive} events
    (joint prob. = 0, e.g., ``individual is male/female'');
    probability of $A$ \emph{or} $B$
    $\equiv \prob(A \cup B) = \prob(A) + \prob(B)$.}
\item{The sum of probabilities of all possible
    outcomes of an observation or
    experiment = 1.0.
    (E.g.: \emph{normalization constants}.)}
\item{
\emph{Conditional probability} of
$A$ \emph{given} $B$, $\prob(A|B)$:
$\prob(A|B) = \prob(A \cap B)/\prob(B)$.
(Compare the \emph{unconditional}
probability of $A$: $\prob(A)=\prob(A|B) + \prob(A|\mbox{not } B)$.)}
\item{If $\prob(A|B)=\prob(A)$,
$A$ is \emph{independent} of $B$.
Independence $\iff$
$\prob(A \cap B)   =  \prob(A) \prob(B)$
(or $\log \prod_i \prob(A_i) = \sum_i \log \prob(A_i)$).
}
\end{enumerate*}

\section*{Probability distributions}

Discrete: probability distribution, cumulative probability distribution.
Continuous: cumulative distribution function, probability \emph{density} function
($p(x) = $ limit of $\prob(x<X<x+\Delta x)/\Delta x$ as $\Delta x \to 0$).

\textbf{Moments}:
Means: $\sum x_i/N = \sum \mbox{count}(x) x/N = \sum p(x) x$ (discrete),
$\int p(x) x \, dx$. Variances: $\sum p(x) (x - \bar x)^2$,
$\int p(x) (x-\bar x)^2 \, dx$. \emph{Higher moments}: skew, kurtosis.

Other descriptors: median, mode.


\section*{Bestiary}

Pretty good summaries on Wikipedia
(\url{http://en.wikipedia.org/wiki/List_of_probability_distributions}).
R help pages.
Books: \cite{bolker_ecological_2008,evans_statistical_2010}, 
Johnson, Kotz, Balakrishnan et al.

Characteristics (discrete vs continuous;
range (positive, bounded, \ldots); symmetric or skewed \ldots)

\begin{itemize*}
  \item Binomial: independent trials from fixed number of trials $N$ with homogeneous
    per-trial probabilities $p$ (\emph{Bernoulli}: $N=1$).
  \item Poisson: sample with independent occurrence,
    homogeneous probability \emph{per unit effort} (lim $N\to\infty$ with $Np$ constant
    of Binomial).
  \item Negative binomial: independent homogeneous Bernoulli trials
    until $m$ successes; \emph{overdispersed} version of Poisson
    (Gamma-Poisson).
  \item Normal (Gaussian): continuous, symmetric; central limit theorem.
  \item Gamma: waiting time (\emph{exponential}: shape=1).
  \item Lognormal: exponential-transformed Gaussian.
  \end{itemize*}
  
\begin{center}
\includegraphics[height=2in,width=2in]{probdisttab_clip}
\end{center}
(Insanely thorough version: \cite{leemis_univariate_2008}.)

\section*{Jensen's inequality}

Bottom line: $E[f(x)] \neq f(E[x])$, unless $f$ is linear
(need new notation for expectation over a particularly probability
density function $p(x)$: $E_p[f(x)] \equiv \int p(x) f(x) \, dx$.
Quantify exactly by calculating the expectation precisely,
or by the \emph{delta} method.

\textbf{Delta method}: $E_p[f(x)] \approx E_p[f(\bar x)] +
E_p[f'(x)|_{x=\bar x} (x-\bar x)] +
1/2 E_p[f''(x)|_{x=\bar x} (x-\bar x)^2] =
f(\bar x) + 1/2 f''(\bar x) \mbox{Var}(x)$.

\small
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%% Monday: probability basics; binomial; negative binomial
%% Tuesday: means, moments, etc.
%% Thursday: