# Coupon collector problem calculator

ldots) (E(W).7) (var(W).99) (P(W ge 10) frac10511296 approx.81096) A box of a certain brand of cereal comes with a special toy. The mean of the number of boxes purchased. This means that the variance is the sum of the variances of the geometric random variables. Let Xk, n be a random variable that denotes the number of batches we need to buy to collect all the n different coupons. Then (bsZ (Z_1, Z_2, ldots, Z_m) is a sequence of independent random variables, and (Z_i) has the geometric distribution on (N with parameter (p_i fracm - i 1m). Am I then right in thinking p_3 would be p_ap_b(p_c p_d) p_ap_c(p_b p_d) p_ap_d(p_b p_c) p_bp_c(p_a p_d) p_bp_d(p_a p_c) p_cp_d(p_a p_b) and p_4 would be p_ap_bp_c(p_d) p_ap_bp_d(p_c) p_ap_cp_d(p_b) p_bp_cp_d(p_c) simplified to 4(p_ap_bp_cp_d). The mean of the sample size.

For fixed (m let (g_k) denote the probability density function of (W_k). Through some research, this appears to be a variation of the coupon collector's problem where each of my "coupons" doesn't have the same probability. (newcommandPmathbbP) (newcommandEmathbbE) (newcommandRmathbbR) (newcommandNmathbbN) (newcommandbsboldsymbol) (newcommandvartextvar) (newcommandcovtextcov) (newcommandcortextcor).

If it helps, I've also made some simulations and estimations, and came up with this table). Let (V_n) denote the number of distinct values in the first (n) selections, for (n. In the coupon collector experiment, vary the parameters and note the shape and location of the mean ( pm ) standard deviation bar. The variance of the sample size. I already know and understand how to calculate the expected value when k1 (that is the classic coupon collector problem but have difficulty calculating EXk, n for. The Probability Density Function, now let's find the distribution of (W_k). From the partial answer from @John: p_1 1 p_2 p_a (p_b p_c p_d) p_b (p_c p_d p_a) p_c (p_d p_a p_b) p_d (p_a p_b p_c). (P(W n) sum_j09 (-1)j binom9j left(frac9-j10right)n-1 for ( n in 10, 11, ldots) (E(W).2897) (var(W) 125.6871) (P(W le 15).04595) Random.

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