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the Expected Value of the Original Variable |
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Let P(x) be the probability density function of a quantity x. Now consider a weighted averagel of P(x); i.e.,
where the weight function w(z) has the properties
All integrals are between −∞ and +∞.
The expected value of x, E_{Π}{x}, according to the probabiliy density function Π is
(In the symbol E_{Π}{x} x stands for the name of the variable rather than a possible value.)
Now consider a change in the variable in the integral on the RHS of the above to x+z=y and hence dx=dy.
Because ∫w(z)dz = 1 and ∫zw(z)dz = 0 the above reduces to
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