From 52eef63fe9b1ed3dca6e4c8a1b11da0f1a081324 Mon Sep 17 00:00:00 2001
From: Julian T <julian@jtle.dk>
Date: Mon, 15 Feb 2021 00:17:29 +0100
Subject: Moved some notes to tex

---
 sem6/prob/m2/noter.md  | 102 -------------------------------------------------
 sem6/prob/m2/noter.tex |  98 +++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 98 insertions(+), 102 deletions(-)
 delete mode 100644 sem6/prob/m2/noter.md
 create mode 100644 sem6/prob/m2/noter.tex

(limited to 'sem6')

diff --git a/sem6/prob/m2/noter.md b/sem6/prob/m2/noter.md
deleted file mode 100644
index f8faec4..0000000
--- a/sem6/prob/m2/noter.md
+++ /dev/null
@@ -1,102 +0,0 @@
-# Noter til probability m2
-
-## Random variables
-
-Her mapper man fra et sample space S til en variabel.
-Her kalder man variablen et stort tal R eller sådan noget.
-
-Derfor er et random variabel egentlig en transformation mellem S og real tal.
-
-*X er en descrete random variable hvis dens range er countable.*
-
-For continues random variables the following is true:
-
-$$
-P(X = x) = 0
-$$
-
-## Functions beskriver ens random variable
-
-### Cumulative Distribution function
-
-Her måler man prob for at ens random er mindre end et bestemt tal.
-
-$$
-F(x) = P(X \leq x)
-$$
-
-Man kan også finde det for en range:
-
-$$
-P(a < X \leq b) = F(b) - F(a)
-$$
-
-Ved discrete random variables vil denne være en slags trappe.
-
-Kan sige at den er *continues from the right* eftersom man har $\leq$ i definition.
-
-### Probability Mass Function
-
-Works only for discrete random variables.
-Is defines as the probability that $X = a$:
-
-$$
-p(a) = P(X = a)
-$$
-
-From here CDF can be found:
-
-$$
-    F(a) = \sum_{all x \leq a} p(a)
-$$
-
-
-
-### Probability Density Function
-
-Her finder man P i et evigt lille interval:
-Is the derivative of the CDF.
-
-$$
-    F(a) = P(X \in (-\infty,a]) = \int_{-\infty}^a f(x) dx \\
-    f(a) = \frac{d}{da} F(a)
-$$
-
-The following must be true:
-
-$$
-    \int_{-\infty}^{\infty} f(x) dx = 1
-$$
-
-## Multiple random variables
-
-Have multiple random variables, which can be or is not correlated.
-Can define the joined CDF:
-
-$$
-    F_{XY}(x,y) = P(X \leq x, Y \leq y)
-$$
-
-One can also find the probability of one of the variables. (The *marginal*)
-
-$$
-    F_X(x) = P(X \leq x) = P(X \leq, Y < \intfy) = F(x, \infty)
-$$
-
-One can not go from marginal to the joined, as they do not contain enough information.
-This is only possible if X and Y are **independent**.
-
-$$
-    F_XY(x,y) = F_X(x) \cdot F_Y(x) \\
-    p(x,y) = p_X(x) \cdot p_Y(y) \\
-    f(x,y) = f_X(x) \cdot f_Y(y)
-$$
-
-### Joined PMF
-
-$$
-    P_{XY}(x,y) = P(X = x, Y = y)
-$$
-
-    vim: spell spelllang=da,en
-
diff --git a/sem6/prob/m2/noter.tex b/sem6/prob/m2/noter.tex
new file mode 100644
index 0000000..3eb2e4f
--- /dev/null
+++ b/sem6/prob/m2/noter.tex
@@ -0,0 +1,98 @@
+\title{Noter til probability m2}
+
+\section{Random Variables}
+Her mapper man fra et sample space S til en variabel.
+Her kalder man variablen et stort tal R eller sådan noget.
+
+Derfor er et random variabel egentlig en transformation mellem S og real tal.
+
+\emph{X er en descrete random variable hvis dens range er countable.}
+
+For continues random variables the following is true:
+
+$$
+P(X = x) = 0
+$$
+
+
+\subsection{Cumulative Distribution Function}
+
+Her måler man prob for at ens random er mindre end et bestemt tal.
+
+$$
+F(x) = P(X \leq x)
+$$
+
+Man kan også finde det for en range:
+
+$$
+P(a < X \leq b) = F(b) - F(a)
+$$
+
+Ved discrete random variables vil denne være en slags trappe.
+
+Kan sige at den er \emph{continues from the right} eftersom man har $\leq$ i definition.
+
+\subsection{Probability Mass Function}
+
+Works only for discrete random variables.
+Is defines as the probability that $X = a$:
+
+$$
+p(a) = P(X = a)
+$$
+
+From here CDF can be found:
+
+$$
+F(a) = \sum_{all x \leq a} p(a)
+$$
+
+
+\subsection{Probability Density Function}
+
+Her finder man P i et evigt lille interval:
+Is the derivative of the CDF.
+
+$$
+F(a) = P(X \in (-\infty,a]) = \int_{-\infty}^a f(x) dx \\
+f(a) = \frac{d}{da} F(a)
+$$
+
+The following must be true:
+
+$$
+\int_{-\infty}^{\infty} f(x) dx = 1
+$$
+
+\subsection{Multiple Random Variables}
+
+Have multiple random variables, which can be or is not correlated.
+Can define the joined CDF:
+
+$$
+F_{XY}(x,y) = P(X \leq x, Y \leq y)
+$$
+
+One can also find the probability of one of the variables. (The \emph{marginal})
+
+$$
+F_X(x) = P(X \leq x) = P(X \leq, Y < \infty) = F(x, \infty)
+$$
+
+One can not go from marginal to the joined, as they do not contain enough information.
+This is only possible if X and Y are \emph{independent}.
+
+$$
+F_{XY}(x,y) = F_X(x) \cdot F_Y(x) \\
+p(x,y) = p_X(x) \cdot p_Y(y) \\
+f(x,y) = f_X(x) \cdot f_Y(y)
+$$
+
+\subsection{Joined PMF}
+
+$$
+P_{XY}(x,y) = P(X = x, Y = y)
+$$
+
+% vim: spell spelllang=da,en
-- 
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