\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: \[
    P(a < X \leq b) = \Phi\left(\frac{b - \mu}{\sigma}\right) - \Phi\left(\frac{a-\mu}{\sigma}\right)
.\]