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\title{Continuous Distributions}
\section{Uniform}
Select a points in the interval $[a, b]$.
\begin{definition}
If a random variable $X \sim Uniform(a, b)$ then its PDF is: \[
f_X(x) = \left\{
\begin{array}{ll}
\frac{1}{b - a} & a < x < b \\
0 & x < a \wedge x > b
\end{array}
\right.
.\]
\end{definition}
The mean value of $X$ is: \[
E[X] = \frac{b+a}{2}
.\]
And the variance is: \[
Var(X) = \frac{(a-b)^{2}}{12}
.\]
\section{Exponential}
Used to model time in between events, or the lifetime of things.
This distribution can be seen as the continuous version of \emph{geometric distribution}.
\begin{definition}
A random variable $X \sim Exponential(\lambda)$ has PDF: \[
f_X(x) = \left\{
\begin{array}{ll}
\lambda \cdot e^{-\lambda x} & x > 0 \\
0
\end{array}
\right.
.\]
\end{definition}
It has the CDF: \[
F_X(x) = \left(1 - e^{-\lambda x}\right) u(x)
.\]
The expected value is: \[
E[X] = \frac{1}{\lambda}
.\]
And the variance is: \[
Var(X) = \frac{1}{\lambda^{2}}
.\]
\begin{theorem}
An exponential random variable is \emph{memoryless} meaning: \[
P(X > x + a | X > a) = P(X > x), \quad \mathrm{for} \; a, x \geq 0
.\]
\end{theorem}
\section{Gaussian or Normal distribution}
The most important distribution is the normal distribution.
\begin{definition}
A random variable $Z \sim N(0, 1)$ has PDF: \[
f_Z(z) = \frac{1}{\sqrt{2 \pi}} \exp\left( - \frac{z^{2}}{2} \right), \quad \mathrm{for all} \; x \in \mathbb{R}
.\]
And $E[Z] = 0, \quad Var(Z) = 1$.
\end{definition}
The scaling of the exponential function is to make sure that the area under the PDF is 1.
The CDF of the normal distribution is denoted with $\Phi(x)$, and is defined with a nasty intergral which has to closed form.
This is often looked up in a table.
\subsection{Scaling and shifting}
One can scale and shift $N(0, 1)$ to make it have other variances and means.
\[
X = \sigma Z + \mu, \quad \mathrm{where} \; \sigma > 0
.\]
And in reverse $Z = \frac{X- \mu}{\sigma}$.
Then $E[X] = \mu$ and $Var(X) = \sigma^{2}$.
Meaning: \[
X \sim N(\mu, \sigma^{2})
.\]
The probability of an interval can be found with: \[
P(a < X \leq b) = \Phi\left(\frac{b - \mu}{\sigma}\right) - \Phi\left(\frac{a-\mu}{\sigma}\right)
.\]
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