Chapter 5.2, Problem 46E

Mean as Center of Gravity. Let X be a discrete random variable with a finite number of possible values, say, x₁, x₂, . . . , x_m. For convenience, set p_k = P(X = x_k), for k = 1, 2, . . . ,m. Think of a horizontal axis as a seesaw and each p_k as a mass placed at point x_k on the seesaw. The center of gravity of these masses is defined to be the point c on the horizontal axis at which a fulcrum could be placed to balance the seesaw.

Relative to the center of gravity, the torque acting on the seesaw by the mass p_k is proportional to the product of that mass with the signed distance of the point x_k from c, that is, to (x_k – c) · p_k. Show that the center of gravity equals the mean of the random variable X. (Hint: To balance, the total torque acting on the seesaw must be 0.)