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Mean Variance Chebyshev

# Mean Variance Chebyshev - IE 111 Fall Semester 2008 Note...

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IE 111 Fall Semester 2008 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 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 hell 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 many many times. How much would you expect to get paid per roll of the die? Answer: 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) 0.333333 0.333333 0.333333 4.333333 2 y 1 2 11 f(y) 0.333333 0.333333 0.333333 4.666667 2 z 1 2 10 f(z) 0.332333 0.333333 0.334333 4.342333 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? Example Consider the following game played over multiple rounds. On round i you flip a coin. 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? E(X 1 ) = (0.5)(\$1)- (0.5)(\$0) = \$0.5 Should you play round 2 given that I won in round 1? E(X 2 ) = (0.5)(\$2)- (0.5)(\$1) = \$0.5 Should you play round 3 given that I won in round 2? E(X 3 ) = (0.5)(\$3)- (0.5)(\$3) = \$0 Should you play round 4 given that I won in round 3? E(X 4 ) = (0.5)(\$4)- (0.5)(\$6) = -\$1 Example Now consider the following random variables Expecte d x -10 2 10 f(x) 0.333333 0.333333 0.333333 0.666667 y 0 0.1 1.9 f(y) 0.333333 0.333333 0.333333 0.666667 z -10 2 11 f(z) 0.333333 0.333333 0.333333 1 Comparing X and Y, many would choose Y (why?) Risk averse. Some gamblers might prefer Y, but such behavior is atypical (suppose the 3 outcomes reflect your retirement savings in \$100,000 units) Now suppose you must choose between Y and Z. You well might choose Y. We except slightly less return in order to greatly reduce risk .

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Mean Variance Chebyshev - IE 111 Fall Semester 2008 Note...

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