res that the PDF mean is (x) = N. The minimum of f is found by solving: (C10) (C11) which yields (C12) and the PDF solution ;=exp(2-1+um) = exp(A - 1)[exp u]"/ (C13) PQmj, with P = exp(2 - 1) and Q = expu To define P, Q, one must then find the two unknown parameters A, u by substituting the result in Eq. (C13) into the two constraints in Eqs. (C4) and (C9): (C14) (C15) which, considering that Q > 0, yields (C16) (C17) Given an arbitrary set of real or integer values m1, m2, Mk, the solution for Q in must be computed numerically, since Eq. (C17) is a transcendental equation of the form (C18) where a;=m;-N.Here, - I will not discuss the conditions for which a real solution Q > 0 may exist in the general case. The only conclusion we can reach is that if such a Appendix C solution Q exists, the PDF defined in Eq. (C13) takes the form: (C19) I show next that this PDF matches the discrete exponential distribution (see Chapter 1), provided we assume that: (a) m1, m2,..., Mk are ordered integers with m1=0,m2=1,...,mi=i 1; (b) k is infinite; (c) u < 0 (for which Q < 1, the condition for the denominator in Eq. (C19) to be finite). With these assumptions, the solution is Q = N/(N + 1) and P = 1/(N + 1), which gives (C20) which is known as the Bose-Einstein (BE) distribution (see Chapter 1). It is shown in particular that in the limit of large means (N >> 1), we have H 2 log N. Incidentally, this is the same entropy as that of a uniform distribution of N discrete events (N an integer). The BE distribution corresponds to the photon statistics of incoherent light, such as those emitted from thermal sources (e.g., candle, light bulb, Sun, stars). It is also characteristic of spontaneous emission in optical amplifiers. Maxwell-Boltzmann distribution solution I consider next another case of interest for the solution in Eq. (C19), where: With these assumptions, all summations in the definitions carry from i = 0 to infinity. We then use the property of the geometrical series get the result pi = (1-Q)Qi from Eq. (C19). To solve for 