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-# Noter til probability m2
-
-## 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.*
-
-For continues random variables the following is true:
-
-$$
-P(X = x) = 0
-$$
-
-## Functions beskriver ens random variable
-
-### 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 *continues from the right* eftersom man har $\leq$ i definition.
-
-### 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:
-
-$$
-    F(a) = \sum_{all x \leq a} p(a)
-$$
-
-
-
-### 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:
-
-$$
-    \int_{-\infty}^{\infty} f(x) dx = 1
-$$
-
-## 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 *marginal*)
-
-$$
-    F_X(x) = P(X \leq x) = P(X \leq, Y < \intfy) = F(x, \infty)
-$$
-
-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**.
-
-$$
-    F_XY(x,y) = F_X(x) \cdot F_Y(x) \\
-    p(x,y) = p_X(x) \cdot p_Y(y) \\
-    f(x,y) = f_X(x) \cdot f_Y(y)
-$$
-
-### Joined PMF
-
-$$
-    P_{XY}(x,y) = P(X = x, Y = y)
-$$
-
-    vim: spell spelllang=da,en
-