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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
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