e, X = {x1 X2, , xk}, where the events take nonnegative real values, I show that the distribution maximizing entropy is the Maxwell-Boltzmann distribution. I then analyze the effect of other additional constraints in the determination of maximum entropy and of the corresponding distributions. Uniform distribution solution By definition, the entropy of the source is: or, with simplified notation: For convenience, I shall use natural logarithms; this does not affect the generality of the following demonstrations. Using the method of Lagrange multipliers, we first define the function with parameter a as: while assuming the constraint: Appendix C This constraint ensures that the sum of all probabilities pi is equal to unity. The task is to minimize f with respect to Pi, namely, to find the solution of: which yields: dy(-pilogi)+) = -logp;-1+1 which yields the solution pj = exp(2-1). - Since 2 is a constant, the distribution is uniform. Substituting this result into the constraint so defined in Eq. (C4) gives so = 1 - which yields the solution exp( - 1) = 1/k (or a = 1 - log k), which finally gives pi = 1/k. The conclusion is that the PDF that maximizes the entropy of any discrete source of k independent events, X = {x1, X2, ..., xk}, is the uniform distribution defined by p(xi) = 1/k. Discrete-exponential (Bose-Einstein) distribution solution We consider next a second problem, which can be formulated as follows: what is the discrete distribution of given mean N that maximizes entropy? We note that if a mean N is specified, the source events must correspond to a discrete set of real values, i.e., X1 = m1, x2 = m2, Xk = mk with Mi # mj for i # j. Whether the values Mi are real or integer, or equally spaced or ordered (i.e., Mi+1 > mi) is not important in the derivation of the general solution to this maximization problem. The solution comes again from the Lagrange-multipliers method, this time with the following function to be minimized: and with the additional constraint: Maximum entropy of discrete sources which ensu