The Formulas of Random Variations And Stochastic Systems 2018.pdf

# The Formulas of Random Variations And Stochastic Systems 2018.pdf

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N N A P A Lim N  N N AB P A B Lim N B  N N AB P AB Lim N  Counting 1. The number of permutations of size k from n different objects. , 1 ... 1 n k k P P n k n n n n k , 2. The number of permutations of size n from a total of n objects where 1 n are of type 1 and so on until k n are of type k . 1 2 1 2 ! , , ,... ! !... ! k k n P n n n n n n n n , 3. Permutation of size k from an infinite supply of each of types (or a supply with at least k of each type). , k P k a a , 4. The number of combinations of size k from n different objects. ! , ! ! ! k n k n n n C C n k k k k n k Axioms of Probability Theory .1. 0 1, Ax P A Formulas of Random Variations & Stochastic Systems

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1 1 .2. 0 if , , a a Si i j Ax P U P s s s i j .3. 1 Ax P S The Axiomatic Formulas P AB P A B P B if 0 P B P AB P B A P A , if 0 P A P AB P A P B A or P B P A B 1 2 3 1 1 1 2 1 1 n n i i i i P A P A P A P A P A A A A A plus ! 1 n other versions. The Theorem of Total Probability Given a random phenomenon where one trial consists in performing a trial of one of m random phenomena 1 2 , , ..., m B B B where 1 P B is the probability of performing a trial of 1 B and so on until m P B is the probability of performing a trial of m B , then for any outcome 1 1 2 2 1 ... m m m i i i P A P B P A B P B P A B P B P A B P B P A B Bayes’ Theorem 1 k k k m i i i P B P A B P B A P B P A B . The Theory of One Random Variable cdf. , , X F x P X x x     pdf. and x X X x X dF x f x F x f u du dx  if x F x is continuous. pdf X f x x P x X x x if X F x is discrete
pmf. , and i X i i X X i x x P x P X x F x P x   b X X X a P a X b F b F a f x dx , if X F x is continuous. P a X b 1 b x i x a P x if X is a discrete random variable X X b X a f x f x a X b f u du ; a x b ; 0 , other i X i x i b X i x a p x p x a X b p x ; i a x b ; 0 , other Leibnitz Rule:  

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• Summer '08
• Staff
• Probability theory, probability density function, FY, Px, Axiomatic Formulas

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