Moved some notes to tex

1 files changed, 22 insertions, 26 deletions

diff --git a/sem6/prob/m2/noter.md b/sem6/prob/m2/noter.tex
index f8faec4..3eb2e4f 100644
--- a/sem6/prob/m2/noter.md
+++ b/sem6/prob/m2/noter.tex
@@ -1,13 +1,12 @@
-# Noter til probability m2
-
-## Random variables
+\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.
 
-*X er en descrete random variable hvis dens range er countable.*
+\emph{X er en descrete random variable hvis dens range er countable.}
 
 For continues random variables the following is true:
 
@@ -15,9 +14,8 @@ $$
 P(X = x) = 0
 $$
 
-## Functions beskriver ens random variable
 
-### Cumulative Distribution function
+\subsection{Cumulative Distribution Function}
 
 Her måler man prob for at ens random er mindre end et bestemt tal.
 
@@ -33,9 +31,9 @@ $$
 
 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.
+Kan sige at den er \emph{continues from the right} eftersom man har $\leq$ i definition.
 
-### Probability Mass Function
+\subsection{Probability Mass Function}
 
 Works only for discrete random variables.
 Is defines as the probability that $X = a$:
@@ -47,56 +45,54 @@ $$
 From here CDF can be found:
 
 $$
-    F(a) = \sum_{all x \leq a} p(a)
+F(a) = \sum_{all x \leq a} p(a)
 $$
 
 
-
-### Probability Density Function
+\subsection{Probability Density Function}
 
 Her finder man P i et evigt lille interval:
 Is the derivative of the CDF.
 
 $$
-    F(a) = P(X \in (-\infty,a]) = \int_{-\infty}^a f(x) dx \\
-    f(a) = \frac{d}{da} F(a)
+F(a) = P(X \in (-\infty,a]) = \int_{-\infty}^a f(x) dx \\
+f(a) = \frac{d}{da} F(a)
 $$
 
 The following must be true:
 
 $$
-    \int_{-\infty}^{\infty} f(x) dx = 1
+\int_{-\infty}^{\infty} f(x) dx = 1
 $$
 
-## Multiple random variables
+\subsection{Multiple Random Variables}
 
 Have multiple random variables, which can be or is not correlated.
 Can define the joined CDF:
 
 $$
-    F_{XY}(x,y) = P(X \leq x, Y \leq y)
+F_{XY}(x,y) = P(X \leq x, Y \leq y)
 $$
 
-One can also find the probability of one of the variables. (The *marginal*)
+One can also find the probability of one of the variables. (The \emph{marginal})
 
 $$
-    F_X(x) = P(X \leq x) = P(X \leq, Y < \intfy) = F(x, \infty)
+F_X(x) = P(X \leq x) = P(X \leq, Y < \infty) = F(x, \infty)
 $$
 
 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 **independent**.
+This is only possible if X and Y are \emph{independent}.
 
 $$
-    F_XY(x,y) = F_X(x) \cdot F_Y(x) \\
-    p(x,y) = p_X(x) \cdot p_Y(y) \\
-    f(x,y) = f_X(x) \cdot f_Y(y)
+F_{XY}(x,y) = F_X(x) \cdot F_Y(x) \\
+p(x,y) = p_X(x) \cdot p_Y(y) \\
+f(x,y) = f_X(x) \cdot f_Y(y)
 $$
 
-### Joined PMF
+\subsection{Joined PMF}
 
 $$
-    P_{XY}(x,y) = P(X = x, Y = y)
+P_{XY}(x,y) = P(X = x, Y = y)
 $$
 
-    vim: spell spelllang=da,en
-
+% vim: spell spelllang=da,en