06-Mean_Variance_Chebyshev

06-Mean_Variance_Chebyshev - IE 111 Fall Semester 2011 Note...

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IE 111 Fall Semester 2011 Note Set #6 Expectation, Variance, and Chebyshev’s Inequality The Mean and Variance of a Random Variable Mean and Variance are two important quantities that describe the behavior of a random variable and PMF. The mean is also equivalently known as the Expected Value. We can denote the mean/expected value of a random variable X using either E(X) or μ X or often just μ . The equation defining the mean for discrete distributions is: E(X) = μ X = all x . x P X (x) The mean can be interpreted as the “center of mass” of a PMF. More importantly it is a measure of the location of the PMF along the x axis. If a PMF "A" has a larger mean than PMF "B", it tends to be located further right. Example 1. Two loaded dice (X, and Y) have the following PMF’s x 1 2 3 4 5 6 --------------------------------------- P X (x) 0.01 0.05 0.1 0.2 0.3 0.34 --------------------------------------- P Y (x) 0.34 0.3 0.2 0.1 0.05 0.01 Find the mean of X and Y. E(X) = (1)(0.01) + (2)(0.05) + (3)(0.1) + (4)(0.2) + (5)(0.3) + (6)(0.34) = 4.75 E(Y) = (1)(0.34) + (2)(0.3) + (3)(0.2) + (4)(0.1) + (5)(0.05) + (6)(0.01) = 2.25 This tells us that “on average” die X will be 4.75 and die Y will be 2.25.
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Interpreting Expected value There are other measures of location including the median and the mode. What is so special about expected value/mean??? Why the heck do they call it expected value? The expected value is not even a possible outcome?!?!?! Suppose we got paid the amount of the die roll, which would you pick? The mean of the two alternatives is a good criterion to choose. Why? This is important! Suppose you repeated this experiment N= many many times. How much would you expect to get paid per roll of the die? Answer: E(X). Or similarly how much money do you expect to make total? Answer: N*E(X). ) ( 1 lim X E n x n i i n = = Consider the following three distributions: Expecte d Median Mode x 1 2 10 f(x) 0.33333 3 0.33333 3 0.33333 3 4.33333 3 2 y 1 2 11 f(y) 0.33333 3 0.33333 3 0.33333 3 4.666667 2 z 1 2 10 f(z) 0.33233 3 0.33333 3 0.33433 3 4.34233 3 2 10 Suppose you got paid in dollars the outcome. Which would you prefer? Expected value is often used as the “basis for rational decision making”. E.g. which investment option would you choose? This is why expected value is so important. Example Consider the following game played over multiple rounds. On round i you flip a coin.
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If it is a head, you win i $. If tails you loose all your money and the game is over You can quit at any time and take home your money. You start with $0. Should you play round 1? i=1 E(X 1 ) = (0.5)($1)- (0.5)($0) = $0.5 Should you play round i=2 given that you won in round 1? E(X
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This note was uploaded on 10/13/2011 for the course IE 111 taught by Professor Storer during the Fall '07 term at Lehigh University .

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06-Mean_Variance_Chebyshev - IE 111 Fall Semester 2011 Note...

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