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authorJulian T <julian@jtle.dk>2021-02-23 12:34:55 +0100
committerJulian T <julian@jtle.dk>2021-02-23 12:34:55 +0100
commit15e3da75a9e78c3ab9b03aa5e7c79b73cf1a2a47 (patch)
tree5281a4abe11a8ffd97627a944e1091f83bd0b8df
parent15a334af05a1f53e37e3ee98021b0032035a7eee (diff)
Add notes for prob m5
-rwxr-xr-xrender.py1
-rw-r--r--sem6/prob/m4/notes.tex4
-rw-r--r--sem6/prob/m5/noter.tex94
3 files changed, 99 insertions, 0 deletions
diff --git a/render.py b/render.py
index 4e189a5..8ae4ed0 100755
--- a/render.py
+++ b/render.py
@@ -14,6 +14,7 @@ tex_template = """\\documentclass[12pt]{article}
\\newtheorem{definition}{Definition}
\\newtheorem{lemma}{Lemma}
+\\newtheorem{theorem}{Theorem}
{% if p is not none %}
\\title{ {{title}} }
diff --git a/sem6/prob/m4/notes.tex b/sem6/prob/m4/notes.tex
index 4a1aea0..227d8bc 100644
--- a/sem6/prob/m4/notes.tex
+++ b/sem6/prob/m4/notes.tex
@@ -143,3 +143,7 @@ The Poisson distribution can be used to approximate binomial distribution.
Two independent Poisson r.v. added together give a poisson distribution with $\lambda = \lambda_1 + \lambda_2$.
\end{lemma}
+
+\begin{lemma}
+ The interarrival between events are a exponential random variable with rate $\lambda$.
+\end{lemma}
diff --git a/sem6/prob/m5/noter.tex b/sem6/prob/m5/noter.tex
new file mode 100644
index 0000000..bce20d4
--- /dev/null
+++ b/sem6/prob/m5/noter.tex
@@ -0,0 +1,94 @@
+\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: \[
+ P(a < X \leq b) = \Phi\left(\frac{b - \mu}{\sigma}\right) - \Phi\left(\frac{a-\mu}{\sigma}\right)
+.\]