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Fenster schliessen

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)
Fenster schliessen

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)
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Independent events

Two events are independent when the following is valid:

 \(P(A\cap B)=P(A)*P(B)\)

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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:


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Bayes' theorem


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what types of random variables do exist?

  1. discrete: number of wet days
  2. continuous (not really!): temperature 
  3. categorial: Head or tail? 
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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. \(P(X \epsilon [a,b])=P(a\leq X\leq b)=F_X(b)-F_X(a)\)
Fenster schliessen

Probability distribution function 

Probability mass function (only for discrete variables!):


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




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