We give exact closed-form expressions for the Kolmogorov and the total variation distances between Poisson, binomial, and negative binomial distributions with different parameters. In the Poisson case, such expressions are related with the Lambert W function.

Abstract:We present a method to obtain both exact values and sharp estimates for the total variation distance between binomial and Poisson distributions with the same mean λ. Concerning exact values, we give an easy formula which holds for moderate sample sizes n, provided that λ is neither close to l + √ l from the left, l = 1, 2, . . ., nor close to m − √ m from the right, m = 2, 3, . . .. Otherwise, a simple efficient algorithm is provided. The zeroes of the second Krawtchouk and Charlier polynomials play a fundamental role.

Abstract-One of the difficulties in calculating the capacity of certain Poisson channels is that H(λ), the entropy of the Poisson distribution with mean λ, is not available in a simple form. In this work we derive upper and lower bounds for H(λ) that are asymptotically tight and easy to compute. The derivation of such bounds involves only simple probabilistic and analytic tools. This complements the asymptotic expansions of Knessl (1998), Jacquet and Szpankowski (1999), and Flajolet (1999). The same method yields tight bounds on the relative entropy D(n, p) between a binomial and a Poisson, thus refining the work of Harremoës and Ruzankin (2004). Bounds on the entropy of the binomial also follow easily.

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