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+\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