[Math] Coupon selecting problem


Suppose that there are N distinct types of coupons and that each time one obtains a coupon, it is, independently of previous selections, equally likely to be any one of the N types. One random variable of interest is T3, the number of coupons that needs to be collected between the acquisitions of the third new type of coupon and the fourth new type of coupon. Find the pmf of T3. Also, find the average number of coupon you need to collect in this case.

I'm not sure how to solve this. If T3 is the number of coupons need to be selected between the third and fourth new coupon. Is it just N-3?

Best Answer

You have just gotten your third different coupon. You draw coupons until you find a fourth one, different from the three you have. Note that these drawing form Bernoulli experiments.

  • What is the probability, in a single draw, to get a new coupon?
  • What random variable is used to model the time it takes to obtain a success in a series of Bernoulli trials?

Added In the same sense,define $T0$ amd $T1$ to be the number of coupons needed to get respectively from $0$ to $1$ coupon and from $1$ to $2$. We start with the easy case $T0$.

Since you have no coupons at hand, you will always draw a new coupon on your next (first) draw. I will do something weird and say $T0$ is a geometric random variable with parameter $N/N=1$.

We go on with $T1$.

You have $1$ of the coupons, so there remains $N-1$ that you need for your collection to be complete. We will compute the probability that you get a new coupon after $k$ new draws, $k=1,2,3,\ldots$. To get a new coupon on the $k$-th draw, you need to first have had $k$ times your first coupon, and then draw a new one. Check that the probabilities this happen is $$ \frac{1}{N^k}\frac{N-1}{N}. $$ This is known has the geometric random variable (linked above) with parameter $p=(N-1)/N$. You should now be able to derive $T2,T3$ and so on.

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