# Cartes-fiches

Cartes-fiches 88 Cartes-fiches 1 Utilisateurs English Université 21.07.2018 / 27.08.2018 Non précisé
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Axioms of probability (Axioms of Kolmogorov)

Probability P : $$\Omega\;\rightarrow\;\ \mathbb{R}$$ (the probability p is a transformation from the event space to the real numbers)

Given events A in an event space $$\Omega$$, i.e., $$A\subset \Omega$$ (A is a subset of Omega; Omega is a superset of A)

1. $$0 \leq P(A) \leq 1$$
2. $$P(\Omega)=1$$
3. given $$A_i\cap A_j =\emptyset$$ for $$i \neq j$$, then $$P(\bigcup_iA_i)=\sum_i P(A_i)$$  (If the intersection of two subsets is zero, then the probability of the union is just the sum of the probabilities of the subsets)

consequences of the Axioms of Kolmogorov

1. $$P(\bar{\bar{A}})=1-P(A)$$
2. $$P(\emptyset)=0$$
3. if A and B are exclusive, then $$P(A\cup B)=P(A)+P(B)$$
4. in general $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$ (additive law of probability)

Independent events

Two events are independent when the following is valid:

$$P(A\cap B)=P(A)*P(B)$$

Conditional probability of two events

The conditional probability of an event A, given an event B is:

$$P(A|B)=P(A\cap B)/P(B)$$

if A and B are independent than:

$$P(A|B)=P(A)$$

Bayes' theorem

$$.\\P(A_j|B)=\frac{P(B|A_j)P(A_j)}{P(B)}$$

what types of random variables do exist?

1. discrete: number of wet days
2. continuous (not really!): temperature
3. categorial: Head or tail?

Cumulative distribution function (CDF)

$$F_X(x)=P(X\leq x)$$ continuous random variables

$$F_X(x)=\sum_{x_i< x}P(X=x_i)$$  discrete random variables

1. $$F_X$$ monotonically increasing ($$0\leq F_X(x)\leq 1$$)
2. $$lim_{x\rightarrow -\infty}F_X(x)=0,\;\;lim_{x\rightarrow \infty}F_X(x)=1$$
3. $$P(X \epsilon [a,b])=P(a\leq X\leq b)=F_X(b)-F_X(a)$$

Probability distribution function

Probability mass function (only for discrete variables!):

$$f_X(x)=P(X=x)$$

Probability density function (PDF, for continous random variables!):

$$f_X(x)=\frac{dF_X(x)}{dx}$$

proberties:

1. $$f_X(x)\geq 0$$
2. $$\int f_X(x)dx=1\;(cont.)\;\;\sum_{X\epsilon \Omega}f_X(x)=1\;(discrete)$$
3. $$P(X\epsilon [a,b])=P(a\leq X\leq b)=F_X(b)-F_X(a)$$
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