# Question: Suppose that X is a continuous random variable with density

Suppose that X is a continuous random variable with density function f. Show that E[|X − a|] is minimized when a is equal to the median of F.

Write

Now break up the integral into the regions where x < a and where x > a, and differentiate.

Write

Now break up the integral into the regions where x < a and where x > a, and differentiate.

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