Lecture16[1]

Lecture16[1] - Lecture 16 Expected Value & Variance...

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Expected Value & Variance Section 5.3 1 STAT 225, Dallas Bateman, Spring 2010 Lecture 16
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Expected Value Question : How do you determine the “value” of a game? Is it better to play Roulette than the Lottery? We are looking for ways of describing random variables. 2 STAT 225, Dallas Bateman, Spring 2010
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Expected Value Definition : Expected Value The expected value of a random variable X with PMF is given by: The expected value is a weighted average of the possible values of X , weighted by the probabilities. 3 STAT 225, Dallas Bateman, Spring 2010 () X px ( ) * ( ) X E X x p x
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Expected Value We may interchangeably use the terms mean , average , expectation , and expected value and the notations E ( X ) or Note: The expected value of a random variable can be understood as the long-run-average value of the random variable in repeated independent trials. If you are playing a game, and X is what you win in the game, then E ( X ) would be your average win if you would play the game many many many times. 4 STAT 225, Dallas Bateman, Spring 2010
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Example #1 Let X be a random variable with PMF: Then 5 STAT 225, Dallas Bateman, Spring 2010 X -1 1/3 0 1/4 1 1/4 2 1/6 (x) p X 1 1 1 1 1 0 1 2 3 4 4 6                         1 4 () EX
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Example #1 Draw the histogram for the PMF above and mark where you think the “balance point” of the histogram would be if the bars were solid metal. 6 STAT 225, Dallas Bateman, Spring 2010 -1 0 1 2 “Balance Point” Should be right About here 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
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E(X) Note: E(X) can be understood as the (physical) “center of gravity” in the histogram, if you consider the probability to be mass (literally). 7 STAT 225, Dallas Bateman, Spring 2010
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Fundamental Expected-Value Formula Instead of E(X) we can also compute the expected value of a function of X: Theorem : Fundamental Expected-Value Formula If X is a discrete random variable with PMF and is any real valued function of X, then 8 STAT 225, Dallas Bateman, Spring 2010 () X px gx [ ( )] ( )* ( ) X E g x g x p x
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Example #2 For the random variable in Example #1 compute the following a) b) c) 9 STAT 225, Dallas Bateman, Spring 2010 2 2 2 2 1 1 1 1 15 ( 1) (0) (1) (2) 3 4 4 6 12                         1.25 1 1 1 1 ( 1 3) (0 3) (1 3) (2 3) 3 4 4 6          
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Lecture16[1] - Lecture 16 Expected Value & Variance...

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