ch7ch8_review - Ch 7 Probability Theory and Stochastic...

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Ch 7 Probability Theory and Stochastic Simulation: Frequentist statistics: - Probability of observing E: ( ) N N E p E - Joint Probability: ( ) ( ) ( ) 1 2 1 2 1 | E E p E p E E p = - Expectation: = = exp 1 exp 1 ) ( N W N W W E ν ν Bayes’ Theorem: ( ) ( ) ( ) ( ) 2 1 2 1 2 1 | | E E p E p E E p E p = - Bayes’ Theorem is general. Definitions : - variance: ( ) ( ) ( ) [ ] ( ) ( ) [ ] 2 2 2 var W E W E W E W E W = = - (X, Y independent, var(X+Y) = var(X) + var(Y)) - standard deviation: ( ) W var = σ - covariance ( ) ( ) [ ] ( ) [ ] { } Y E Y X E X E Y X = , cov , for two random variables X and Y - covariance matrix Important Probability Distributions Definitions: - Discrete random variable o For { } m i X X X X ,..., , 2 1 = o ( ) = i X N number of observations of Xi o T is the total number of observations ( ) T X N M i j = = 1 o Probability is definied by: ( ) ( ) T X N X P i i = o Normalization is defined by ( ) 1 1 = = M i j X P - Continuous random variable o This is just the continuous version of the above, defined by integrals instead of limits, differentials instead of increments o Normalization condition: ( ) = hi x lo x dx x p 1 o Expectation ( ) ( ) = = hi x lo x dx x xp x x E - Cumulative Probability distribution o Basis of RAND in matlab
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o ( ) ( ) u dx x p X F x lo x M = = ' ' o u is defined as uniformly distributed 1 0 u Bernoulli trials o Concept that observed error is the net sum of many small random errors Random Walk Problem - key point: independence of coin tosses - Main results: 0 = x 2 2 nl x = Binomial Distribution - probability distribution: ( ) ( ) ( ) H n n H H n H H H p p n n n n P = 1 , - binomial coefficient: ( ) ! ! ! H H H n n n n n n = - BINORND Matlab to generate random number distributed using binomial distribution Gaussian (Normal) Distribution - Take binomial distribution, change into probability of observing net displacement after n steps of length l o ( ) ( ) ( ) n l x n l x n n l n x p + = 2 1 2 / !
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