in showing a probabilistic bound on the maximal overhang like rm Pr(hat N leq k) geq mathcal F(k,b,N,I(mathbf p as a function expression mathcal F, which incorporate an information theoretic measure mathcal I(mathbf.g. The "traditional coupon collecting problem" can be summarized as follows: Assume we have a variable numbers n of draws from a non-depleting set of coupons (urn with coupons distributed as mathbf p). Therefore, ti has geometric distribution with expectation 1/. E(50) 50(1 1/2 1/3. In the general case of a nonuniform probability distribution, according to Philippe Flajolet, 3 E(T)int _0infty big (1-prod _i1n(1-e-p_it)big )dt. Old description: Topic: Multinomial proxy variables: Bound on probability of their sums.
Graph of number of coupons, n vs the expected number of tries (i.e., time) needed to collect them all, E t in probability theory, the coupon collector's problem describes the "collect all coupons and win" contests. Thanks @kjetil-b-halvorsen for your comment. Dawkins, Brian (1991 "Siobhan's problem: the coupon collector revisited The American Statistician, 45 (1 7682, doi :.2307/2685247, jstor 2685247. 8587, 191, isbn, MR 1265713. Common generalization, also due to Erds and Rnyi: as.displaystyle operatorname P bigl (T_m nlog n(m-1)nlog log ncnbigr )to e-e-c m-1)!, textas nto infty. At least some bound dependend on maybe max_i p_i and min_i p_i would be helpful. Is the EulerMascheroni constant. Think of T and ti as random variables. The derived bounds require much less computation than the exact formula.