As in Exercise 37, a 4 m long wooden board has a fulcrum placed 1 m from the left end. A 20 kg child sits at the right end of the board (at position x = 3) and an 80 kg adult sits at the left end (at position x = -1). Suppose that the board has density of 5 kg/m. Find the center of mass. Who will go up?
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