chapter 4mathstat
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Chapter 4mathstatTRANSCRIPT
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Mathematical Expectation
31.10.2013Dr. Bilge Eriş
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“Mathematical Expectation”• arose in connection with games of chance• Amount the player stands to win*probability
that he will win– one of the 10.000 tickets with a prize of $4800– one of the 10.000 tickets with a prize of $1200– one of the 10.000 tickets with a prize of $400
• On average win;
– Sum of the products obtained by multiplying each amount by the corresponding probability
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• if there is a second prize worth 1200$ and third worth 400$– On average 4800+1200+400/10.000=0.64$ per
ticket
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4.2 THE EXPECTED VALUE OF A RANDOM VARIABLE
• The amount won: Random variable• Sum of the products: Mathematical
expectation• Refer mathematical expectation of a random
variable as expected value
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We may be expected in expected values of random variables related with x
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Calculating expected values from other known expectations
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Possible to extend mathematical expectation
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4.3 MOMENTS• Mathematical expectations called moments of
a random variable
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When r=1, we have
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Other special moments
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4.4 CHEBYSHEV’S THEOREM
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• P(X will take on value within 2 sd. of the mean)=3/4 at least
• P(X will take on value within 3 sd. of the mean)=8/9 at least
• Only when the distribution is known, we can calculate the exact probability
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4.5 MOMENT GENERATING FUNCTIONS
• Alternative procedure to calculate moments
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Basic theorems
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4.6 PRODUCT MOMENTS
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• Indepence implies zero covariance but zero covariance does NOT imply independence!
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Generalization of product moments
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4.7 MOMENTS OF LINEAR COMBINATIONS OF RANDOM VARIABLES
• Derive mean, variance of linear combinations
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4.8 CONDITIONAL EXPECTATION
• Likewise the conditional distributions
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