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C oupon collector’s problem

C oupon collector’s problem. CS658 Po- Ching Liu. Question . A person trying to collect each of b different coupons must acquire approximately x randomly obtained coupons in order to succeed. find x . Balls and bins. 1 st question:

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C oupon collector’s problem

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  1. Coupon collector’s problem CS658 Po-Ching Liu

  2. Question • A person trying to collect each of b different coupons must acquire approximately x randomly obtained coupons in order to succeed. • find x

  3. Balls and bins • 1st question: If n identical balls are tossed randomly into b bins, thenhow many balls willfall in a given bin? • Ans:

  4. Second • Second question: Find the expected number of balls we need to toss (one ball for each toss) until a given bin contains a ball. The number of bins = b • Ans: E = b

  5. proof • p=pro(success) = 1/b=1-q • q=pro(withoutsuccess)=1-1/b=1- p • E[# of tosses until a given bin contains a ball]

  6. Third • the third question: Find the expected number of balls we need to toss (one ball for each toss) until every bin contains at least one ball? The number of bins = b • Ans: E = b(lnb+O(1))

  7. proof • There are b stages • The ith stage consists of the tosses after the (i-1)th hit until the ith hit. • how many balls do we have to toss in order to move from the (i-1) th stage to the ith stage  there are (i-1) bins that contain balls and b-(i-1) empty bins. for each toss in the ith stage, the probability of obtaining a hit is (b-i+1)/b.

  8. proof • E[# of tosses in the ith stage]= • E[total # of tosses in the 1st to the bth stage] =

  9. proof • We find the two following questions are the same: • Q:Find the expected number x of balls we need to toss (one ball for each toss) until every bin contains at least one ball? The number of bins = b • Q: A person trying to collect each of b different coupons must acquire approximately x randomly obtained coupons in order to succeed.

  10. Code by java compile: javac CouponCollector.java Run: java CouponCollector • public class CouponCollector { • public static void main(String[] args) { • int N = 50; // number of different card types • boolean[] found = new boolean[N]; • // found[i] = true ==> if card i has been collected , false ==> the new type of card • intcardcnt = 0; • // total number of cards collected • intvalcnt = 0; • // number of distinct cards • // repeatedly choose a random card and check whether it's a new one • while (valcnt < N) { • intval = (int) (Math.random() * N); • // random card between 0 and N-1 • cardcnt++; • // we collected one more card ==>total number +1 • if (!found[val]) valcnt++; • // it's a new card type • found[val] = true; • // update found[] • } • // print the total number of cards collected • System.out.println(cardcnt); • } • }

  11. references • Thomas H. Cormen,Charles E. Leiserson, Ronald L. Rivest, Clifford Stein(1990), Introduction to Algorithms, The MIT Press, Cambridge, Massachusetts London, England, pp. 109-110. • http://www-stat.stanford.edu/~susan/surprise/Collector.html

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