Question

The maximum entropy distribution is Gaussian with two constraints. Use the Lagrange multiplier method to prove that the probability distribution p_i that maximizes the entropy for die rolls, subject to a constant value of the second moment 〈i₂〉, is a Gaussian function. Use ε_i = i. Two constraints:

Answer 1

The maximum entropy distribution is Gaussian (with two constraints) show the second moment is given maximize the entropy subject to two constraints sigma^t_i = 1 P_p = 1, sigma^t_i = 1 = i^2 P_p = < i^2 > = infinity we have - 1 - ln P_p^* - lambda - beta^2_p = 0 Rightarrow P^*_p = e^- 1 - lambda beta^2_i rightarrow (1) Dividing both sides of equation (1) by sigma^t_i = 1 e^- 1- lambda - beta^2_p gives P^*_p = e^- P^2_p/sigma^t_i = 1 e^- beta^2_p Therefore which is a Gaussian function of l.