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)
consequences of the Axioms of Kolmogorov
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?
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
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: