statistics for atmospheric science
statistic for atmospheric science
statistic for atmospheric science
Fichier Détails
Cartes-fiches | 88 |
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Langue | English |
Catégorie | Mathématiques |
Niveau | Université |
Crée / Actualisé | 21.07.2018 / 27.08.2018 |
Attribution de licence | 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)
- \(0 \leq P(A) \leq 1\)
- \(P(\Omega)=1\)
- 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
- \(P(\bar{\bar{A}})=1-P(A)\)
- \(P(\emptyset)=0\)
- if A and B are exclusive, then \(P(A\cup B)=P(A)+P(B)\)
- 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?
- discrete: number of wet days
- continuous (not really!): temperature
- 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
- \(F_X\) monotonically increasing (\(0\leq F_X(x)\leq 1\))
- \(lim_{x\rightarrow -\infty}F_X(x)=0,\;\;lim_{x\rightarrow \infty}F_X(x)=1\)
- \(P(X \epsilon [a,b])=P(a\leq X\leq b)=F_X(b)-F_X(a)\)
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:
- \(f_X(x)\geq 0\)
- \(\int f_X(x)dx=1\;(cont.)\;\;\sum_{X\epsilon \Omega}f_X(x)=1\;(discrete)\)
- \(P(X\epsilon [a,b])=P(a\leq X\leq b)=F_X(b)-F_X(a)\)