Screenshot 2018-09-30 at 00.25.43.png - What the above demonstrates is two basic rules for expectations operators regardless of whether random variables

Screenshot 2018-09-30 at 00.25.43.png - What the above...

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Unformatted text preview: What the above demonstrates is two basic rules for expectations operators regardless of whether random variables are discrete or continuous: 1) E(a) = a “The expected value of a constant is a constant” 2) E(bX) = bE(X) “The expected value of a constant times a random variable is equal to the constant times the expected value of the random variable” More generally let X Y and Z be three random variables eanch with their own density functions and let W = aX + bY + cZ, then E(W) = aE(X) + bE(Y) + cE(Z). 2. g(X) = (Ii-EGO)i i = 2, .. Functions of this form yield “i’th moments about the mean”. The metric here is the i’th power of that of the original distribution so that if f(x) relates to incomes measured in $US then the i’th moment about the mean is measured in ($US)i. Sometimes the i’th root of g(X) is employed since it yields a measure of the appropriate characteristic in terms of the original units of the distribution. Furthermore to make distributions measured under different metrics comparable the filnction g(X) deflated by the appropriate power of E(X) (provided it is not 0) is considered, providing a metric free comparator. The second moment (i = 2) is of particular interest1 since as the variance (frequently represented as 02 or V(X), its square root 0 is referred to as the standard deviation) it provides a measure of how spread out a distribution is. Of particular interest here is ...
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