\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}

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

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