# Example 2.2.6 The expected value of a random variable has another property, one that we can think…

Example 2.2.6 The expected value of a random variable has another property, one that we can think of as relating to the interpretation of E X as a good guess at a value of X. Suppose we measure the distance between a random variable X and a constant 6 by (X —0z. The closer b is to X, the smaller this quantity is. We can now determine the value of 6 that minimizes E(X — 6)2 and, hence, will provide us with a good predictor of X. (Note that it does no good to look for a value of 6 that minimizes (X — b)2, since the answer would depend on X, making it a useless predictor of X.) We could proceed with the minimization of E(X — 6)2 by using calculus, but there is a simpler method. (See Exercise 2.19 for a calculus-based proof.) Using the belief that there is something special about E X, we write

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