1. Problem10.15LetXbeacontinuousrandomvariablewithprobabilitydensity f(x)andfiniteexpectedvalueE(X).

10.1 Concept of probability density 295

(a) What constant c minimizes E[(X − c)2] and what is the minimal value of E[(X −c)2]?

(b) Prove that E(|X − c|) is minimal if c is chosen equal to the median of X, where the median of X is any value m for which P(X ≤m)= P(X ≥m)= 1 2

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