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\title{Noter til probability m2}
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:
$$
P(X = x) = 0
$$
\section{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.
\section{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:
$$
F(a) = \sum_{all x \leq a} p(a)
$$
\section{Probability Density Function}
Her finder man P i et evigt lille interval:
In the following formula PDF is $f$.
\begin{equation*}
\begin{split}
F(a) = P(X \in (-\infty,a]) = \int_{-\infty}^a f(x) dx \\
f(a) = \frac{d}{da} F(a)
\end{split}
\end{equation*}
The following must be true:
$$
\int_{-\infty}^{\infty} f(x) dx = 1
$$
\section{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.
However if two random variables, and $A$ and $B$ are two sets of real numbers:
\[
P(X \in A, Y \in B) = P(X \in A) P(Y \in B)\,.
\]
% This is only possible if X and Y are \emph{independent}.
% \begin{align*}
% 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)
% \end{align*}
\section{Conditional PDF}
If $X$ and $Y$ have a joint PDF, then the conditional PDF of X given that $Y=y$ is
\[
F_{X|Y}(x|y) = \frac {f(x, y)} {f_Y(y)}
\]
There is also one for PMF not listed here.
\section{Joined PMF}
$$
P_{XY}(x,y) = P(X = x, Y = y)
$$
% vim: spell spelllang=da,en
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