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lecture03 - 16.322 Stochastic Estimation and Control, Fall...

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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Lecture 3 Last time: Use of Bayes’ rule to find the probability of each outcome in a set of ( | 1 , PA E E 2 ,... E ) k n In the special case when E’s are conditionally independent (though they all depend on the alternative, A k ), ( | 1 , ( | 1 ... ( | k n 2 ,... E ) = PA E E n 1 ) PE A ) . k n ( | 1 ... ( | ) ) i n 1 n i i This is easy to do and can be done recursively. ( | () (| k 1 k Take E 1 : PA E ) = PA PE A ) k 1 ()( | PA ) i 1 i i ( | ( | ) ( | k 1 2 k Take E 2 : PA EE ) = PA E PE A ) k 12 (|) ( | ) i 1 2 i i Î Recursive application of Bayes’ rule. Random Variables Start with a conceptual random experiment: Î Continuous-valued random variable Î Discrete-valued random variable The random variable is defined to be a function of the outcomes of the random experiment. Example: X = the number of spots on the top face of a die. How to characterize a random variable? Page 1 of 7
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Probability distribution function (or cumulative probability function) Fx ( () = PX x ) , where x is the argument and X is the random variable name . Properties: F ( −∞ ) = 0 F () 1 ∞= ( 0 Fx ) 1 Fb ( Fa ) , i f b> a Continuous random variable Discrete random variable An alternate way to characterize random variables is the probability density function : dF x fx = dx () 0 , ≥∀ x x f x ) = F ( + f u d u −∞ x fud u = −∞ Page 2 of 7
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16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde ( ( ( Pa < X b ) = PX b ) a ) () Fa ) = Fb ( b a () f x dx f x dx = −∞ −∞ b fxd x = a F ∞= f u d u = 1 −∞ The density function is always a non-negative function, which integrates over the full range to 1.
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This note was uploaded on 11/07/2011 for the course AERO 16.322 taught by Professor Wallacevandervelde during the Fall '04 term at MIT.

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lecture03 - 16.322 Stochastic Estimation and Control, Fall...

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