Lect7_2700_s09 - ENGRD 2700 Basic Engineering Probability...

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Expectation Rules Variance of X Chebychev Mass functions Title Page JJ II J I Page 1 of 31 Go Back Full Screen Close Quit ENGRD 2700 Basic Engineering Probability and Statistics Lecture 7: Expectation, Variance, Distributions David S. Matteson School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA [email protected] February 9, 2009
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Expectation Rules Variance of X Chebychev Mass functions Title Page JJ II J I Page 2 of 31 Go Back Full Screen Close Quit 1. Expectation of a discrete RV Suppose ( S, A , P ) is a probability model and let X be a random vari- able. Imagine Do the experiment. Get an outcome s S . The random variable X gives us a number X ( s ). What number do we EXPECT before we do the experiment? Definition: The expectation E ( X ) of the discrete random variable X is E ( X ) := X x ∈{ possible values } xP [ X = x ] = X x ∈{ possible values } xp X ( x ) . In words: To compute E ( X ): Take a possible value x and weight it by the probability P [ X = x ] the rv X takes that value. Do this for each possible value and sum.
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Expectation Rules Variance of X Chebychev Mass functions Title Page JJ II J I Page 3 of 31 Go Back Full Screen Close Quit Remarks: E ( X ) is sometimes called the first moment in analogy with physics. If both S and X are discrete, this is also equal to E ( X ) = X s S X ( s ) P ( { s } ) since by regrouping, the last sum can also be written as X x ∈{ possible values } X s : X ( s )= x X ( s ) P ( { s } ) = X x ∈{ possible values } X s : X ( s )= x x P ( { s } ) = X x ∈{ possible values } x P [ X = x ] . As with the arithmetic mean of a sample of numbers, the expec- tation need not be a possible value.
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Expectation Rules Variance of X Chebychev Mass functions Title Page JJ II J I Page 4 of 31 Go Back Full Screen Close Quit Example 1: X -1 1 p X ( x ) 1/2 1/2 In this example, the possible values are {- 1 , 1 } but E ( X ) = ( - 1) 1 2 + (1) 1 2 = 0 , and 0 is not a possible value. Example 2: Throw a die: X 1 2 3 4 5 6 p X ( x ) 1/6 1/6 1/6 1/6 1/6 1/6 Now E ( X ) = 6 i =1 i 6 = 21 6 = 3 . 5 / ∈ { 1 , 2 , 3 , 4 , 5 , 6 } .
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Expectation Rules Variance of X Chebychev Mass functions Title Page JJ II J I Page 5 of 31 Go Back Full Screen Close Quit Relative frequency interpretation Experiment: Observe X a large number of times, say M . Suppose the realizations give numbers z 1 , z 2 , . . . , z M . Average these numbers: average =: z 1 + z 2 + · · · + z M M = X i x i × (# z ’s = x i ) M (regroup sum) X i x i p X ( x i ) . (relative freq prob )
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Expectation Rules Variance of X Chebychev Mass functions Title Page JJ II J I Page 6 of 31 Go Back Full Screen Close Quit Another Example: Mean of binomial RV Suppose N has a binomial mass function. Notation: N b ( k ; n, p ) . Then E ( N ) = n X k =0 k n k p k q n - k = n X k =1 k n ! k !( n - k )! p k q n - k = pn n X k =1 ( n - 1)! ( k - 1)!( n - k )! p k - 1 q n - k = pn n X k =1 n - 1 k - 1 p k - 1 q n - k and changing dummy index from k to j = k - 1 gives = pn n - 1 X j =0 n - 1 j p j q n - j - 1 = pn n - 1 X j =0 n - 1 j p j q n - 1 - j = pn n - 1 X j =0 b ( j ; n - 1 , p ) = pn × 1 = pn.
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Expectation Rules Variance of X Chebychev Mass functions Title Page JJ II J I Page 7 of 31 Go Back Full Screen Close Quit The Geometric Distribution.
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