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\title{Special Probability Distributions}
TODO Look at discrete distributions.
\section{Moment Generations}
\emph{These are left over from the last lecture.}
Two random variables with the same $E[X]$ and $Var(X)$ and variance are not the same.
Instead one can calculate the expantance of a higher order of random variable.
\begin{definition}
The $n$'th moment of a r.v. is defined as: \[
E[X^n] = \sum_{i} x_{i}^{n} p(x_i)
.\]
\end{definition}
% TODO Describe how to extract a moment
\begin{definition}
The \emph{Moment Generation Function} for a r.v. variable is defined as: \[
\varphi(t) = E[e^{t X}] = \sum_{i} e^{t x_i} p(x_i)
.\]
The continues version is given by: \[
\varphi(t) = E[e^{t X}] = \int_{-\infty}^{\infty} e^{tx} f(x) \,dx
.\]
\end{definition}
From this function one can generate all moments of the random variable X.
The variance can be calculated from the first two moments.
\begin{lemma}
If two r.v. have the same moments they can be said to be the same.
\end{lemma}
\section{Discrete Distributions}
These are all covered nicely in the book, in section 3.1.5.
\subsection{Bernoulli}
\begin{definition}
If a random variable is \emph{Bernoulli} with probability $p$, its PMF is: \[
P_X(x) = \left\{
\begin{array}{ll}
p & \mathrm{for} \: x = 1 \\
1 - p & \mathrm{for} \: x = 0 \\
0 & \mathrm{otherwise} \\
\end{array}
\right.
.\]
\end{definition}
The Bernoulli random variable can also be called the \emph{Indicator} random variable.
Because either event $A$ occurs or not.
\subsection{Geometric}
Is a series of independent Bernoulli tails, such as the number of coin tosses before a heads occurs.
\begin{definition}
If X is \emph{geometric} with parameter $p$ its PMF is: \[
P_X(k) = \left\{
\begin{array}{ll}
p(1-p)^{k-1} & \mathrm{for} \: k = 1,2,3,... \\
0 & \mathrm{otherwise}
\end{array}
\right.
.\]
where $0 < p < 1$.
\end{definition}
\subsection{Binomial}
Suppose a coin toss with $P(H) = p$.
If the coin is tossed $n$ times $X$ defines the number of heads that are observed.
\begin{definition}
If $X \sim Binomial(n,p)$, X is said to be \emph{binomial} and its PMF is: \[
P_X(k) = \left\{
\begin{array}{ll}
\binom{n}{k} p^k (1 - p)^{n-k} & \mathrm{for} \: k = 0,1,2,...,n \\
0 & \mathrm{otherwise}
\end{array}
\right.
.\]
where $0 < p < 1$.
\end{definition}
\subsection{Pascal}
Is also called \emph{Negative binomial} and describes the number of trails before $m$ successes.
\begin{definition}
If $X \sim Pascal(m,p)$ its PMF is: \[
P_X(k) = \left\{
\begin{array}{ll}
\binom{k-1}{m-1} p^m (1-p)^{k-m} & \mathrm{for} \: k=m,m+1,m+2,... \\
0 & \mathrm{otherwise}
\end{array}
\right.
.\]
where $0 < p < 1$.
\end{definition}
\subsection{Hyper geometric}
Suppose that a bag contains $b$ blue and $r$ red marbles, and $k \leq b + r$ marbles are chosen.
Then $X$ is the number of chosen blue marbles.
\begin{definition}
If $X \sim Hypergeometric(b,r,k)$ its PMF is: \[
P_X(k) = \left\{
\begin{array}{ll}
\frac{\binom{b}{x} \binom{r}{k-x}}{\binom{b+r}{k}} & \mathrm{for} \: x \in R_X \\
0 & \mathrm{otherwise}
\end{array}
\right.
.\]
where $R_X = \{\max(0, k-r), \max(0,k-r)+1,...,\min(k,b)\}$.
\end{definition}
\subsection{Poisson}
Can be used very well to model random variables in nature.
\begin{definition}
A random variable with values 0,2,3,... can be said to be Poisson with parameter $\lambda > 0$, with PMF: \[
P(X = i) = e^{-\lambda} \frac{\lambda^i}{i!}
.\]
\end{definition}
The expected value is: $
E[X] = \lambda
$
And the variance is: $
Var(X) = \lambda
$
The Poisson distribution can be used to approximate binomial distribution.
\begin{lemma}
Two independent Poisson r.v. added together give a poisson distribution with $\lambda = \lambda_1 + \lambda_2$.
\end{lemma}
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