Find the center of mass of a bar with mass density p(x) = x2(2 - x) for 0 ‰¤ x ‰¤ 2. Is it to the left or the right of the center of the bar at x = 1 ? Is it at the point where p(x) takes on its maximum?
The relation between the mathematical expectation and the center of mass in physics also holds for continuous distributions. Mass density acts like probability density (after the mass density has been divided by the total density). For example, suppose the density of a 1 m long bar is p(x) = 4x kg/m. Then the total mass is

Dividing p(x) by this total mass of 2 gives a density f(x) = 2x, which has integral 1, like a p.d.f. The expectation of a random variable X with this p.d.f. is

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2x32 31-3 E(X) = xf(x) dx = 2x2