HO10-Equal_Areas - MS&E 352 Decision Analysis II Handout#10...

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MS&E 352 Handout #10 Decision Analysis II February 5 th , 2009 _________________________________________________________________________________ ____________________________________________________________________________ 02/11/08 page 1 of 4 Equal Areas The Idea Behind "Equal Areas" In discretizing a continuous probability distribution, the "equal areas" method is a convenient way of converting a continuous distribution of values for a distinction into a handful of select values having discrete probabilities associated with them. This discretization method can be performed by graphically examining the cumulative distribution function, and thus it is easier to explain to decision-makers. The underlying theory behind the “equal areas” method is not difficult to comprehend, and shall be explained in this handout. The conditional mean value of a random variable having probability density function f , given that the random variable takes on a value in the range [ a , b ], where a < b , is given by (1) X a , b = E( X | X [ a , b ] ) = x f (x) dx a b f (x) dx a b = 1 F ( b ) - F ( a ) x f (x) dx a b Note that when a = -∞ and b = + , the above formula simply reduces to the definition of the unconditional mean of a random variable: E( X | X [- , + ] ) = x f (x) dx - + f (x) dx - + = x f (x) dx - + = E(X) since f (x) dx - + = 1 Consider the graph of the cumulative distribution function F (x) below: x 1 x 2 p 1 p 2 A1 A3 A2 F(x) x y
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MS&E 352 Handout #10
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