Another way to derive the bell curve is as the maximum entropy distribution with known mean and variance. The derivation does require one trick that is usually

Another way to derive the bell curve is as the maximum entropy distribution with known mean and variance. The derivation does require one trick that is usually not taught in high school. You know how you can maximize a function by setting its derivative to zero? Well, there is a generalization of that to functions of multiple variables: you set the gradient to zero. But there is yet another generalization to that for functions of continuously infinite number of variables! That's called the calculus of variations. If we want to maximize the entropy over all possible probability distributions over the real numbers, then we have a continuously infinite number of variables, since for each real number x, we need to assign a probability mass p(x). Here is the derivation for those interested: Let m be the mean we want and v be the variance we want, and let p(x) be the probability density function. We want to maximize the entropy S = int(p(x)log(p(x))) given the constraints that: int(p(x)) = 1 // total probability should sum to 1 E(X) = int(x*p(x)) = m // mean should be m E((X-m)^2) = int((x-m)^2*p(x)) = v // variance should be v This problem can be solved with calculus of variations with Lagrange multipliers. The Lagrangian is all the terms inside the integrals summed up: L = p*log(p) + a*p + b*x*p + c*(x-m)^2*p Calculus of variations tells us that for the optimal p, we have: dL/dp = 0 This is the analogue of setting the derivative to zero for maximizing a function. Calculating that we get: a + bx + c(x-m)^2 + log p + 1 = 0 rewriting: p = e^-(a + bx + c(x-m)^2 + 1) We have the general form of the Bell curve. Now we just need to find the right values for a,b,c to satisfy the 3 constraints. As special cases of this we have: 1) The probability distribution with maximum entropy without constraints is the uniform distribution. 2) The probability distribution with maximum entropy with only the constraint that the mean is m, is the exponential distribution. In general if you want to find maximum entropy distribution with known expectation values E(f(X)) and E(g(X)), then that is of this form: p(x) = C*e^(a*f(x) + b*g(x)) This gives an indication why so many distributions have this form. For example the entire exponential family can also be derived as maximum entropy distributions with certain known expectation values. Of course this derivation is only a good motivation for the bell curve if you already cared about entropy, mean and variance.