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Unformatted text preview: 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 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. 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. 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: X1 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 } . 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 ) 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 ) ....
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 Spring '05
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 Probability theory, Chebychev Mass functions

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