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Crée / Actualisé 21.07.2018 / 27.08.2018
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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)
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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:

\(P(A|B)=P(A)\)

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Bayes' theorem

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

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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. \(P(X \epsilon [a,b])=P(a\leq X\leq b)=F_X(b)-F_X(a)\)
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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. \(P(X\epsilon [a,b])=P(a\leq X\leq b)=F_X(b)-F_X(a)\)