Question: Let X be a random variable of the continuous type that has pdf f(x). If m is the unique median of the distribution of X

Let X be a random variable of the continuous type that has pdf f(x). If m is the unique median of the distribution of X and b is a real constant, show that

E (X b}) = E(X m)) + 2 [*(b x) (x) dx,

provided that the expectations exist. For what value of b is E(|X −b|) a minimum?

E (X b}) = E(X m)) + 2 [*(b x) (x) dx, - -

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