Screenshot 2018-09-30 at 00.25.37.png - Two g functions of special interest 1 g(X =X Obviously this yields E(X the expected value of the random variable

# Screenshot 2018-09-30 at 00.25.37.png - Two g functions of...

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Unformatted text preview: Two g( ) functions of special interest. 1. g(X) =X. Obviously this yields E(X), the expected value of the random variable itself (frequently referred to as the mean and represented by the character p), which is a constant providing a measure of where the centre of the distribution is located. The metric here is the same as that of the random variable so that, if f(x) is an income distribution measured in \$US, then its location will be in terms of a \$US value. The usefulness of the linearity property of the expectations operator can be seen by letting g(X) = a + bX where a and b are ﬁxed constants. Taking expectations of this g(X) yields the expected value of a linear function of X which, following the respective rules of summation and integration, can be shown to be the same linear function of the expected value of X as follows: E(a+bX) = )3 (a+bx,)j(x,) = a E ﬁx9+b Z xﬂxg = a+bE(X) all possible at, all possible 1:, all possible at, E(a+bX) = f (a + bx))‘(x)dx = a fﬂxﬂx + b mxﬂx)dx = a+bE(X) ...
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