\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}

$$
P_{XY}(x,y) = P(X = x, Y = y)
$$

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