Question: Let X be a continuous random variable with density function f . Recall that the median of the distribution of X is the real number

Let X be a continuous random variable with density function f . Recall that the median of the distribution of X is the real number m such that P(X ≤ m) = P(X ≥ m).

Show that, if the distribution of X is symmetric around a point a (see Exercise 21 of Section 6.1), and the expectation E(X) exists, then m = E(X) = a.

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