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April 30

Calculating Expected Value

Say that I randomly choose k balls out of n balls with repetitions, until the first time I get some ball that I've already chosen in the past (I can choose it again, since tere're repetitions). Let X denote the number of choices until the first time when I choose some ball that I've already chosen. What is the expected value of X? I thought that $\mathrm{Pr}(X=k+1)=\frac{(\genfrac{}{}{0ex}{}{n}{k})}{n^k}\cdot \frac{k}{n}$ (because I need to choose a subset of k distinct balls, and then, to choose one of thes balls), but this leads to expected value < 1, which does not make sense to me. What is the correct expected value of X? Thanks in advance! 80.246.136.227 (talk) 06:12, 30 April 2016 (UTC)

I'm not sure there can even be an analytic formula for this. If k > n/2, the exact number of repetitions is 2, and so the expected value is 2. For k = n/2, the exact number of repetitions is either 2 or 3, the latter being very unlikely, so the expected value is slightly greater than 2. So it seems that there's a sort of kink point in the expected value function at k = n/2. Likewise, for k > n/3 the actual number of repetitions can be no more than 3, but for k = n/3 there is a slight chance of as many as 4 repetitions, so it seems that there's a kind of kink point there too. That's why I doubt there's an analytic function for the expected value. Loraof (talk) 15:49, 30 April 2016 (UTC)

For what it's worth, I've worked out the k=1 case (derivation omitted here), which is far simpler than other cases. We have

$Pr(X=x)=\frac{x-1}{n^{x-1}}\prod _{j=2}^{x-1}(n-(x-j))$

and

$E(X)=\sum _{x=2}^{n+1}(\frac{x(x-1)}{n^{x-1}}\prod _{j=2}^{x-1}(n-(x-j))).$

Loraof (talk) 21:58, 30 April 2016 (UTC)

And the other relatively simple case has k=n/2. Then X=3 if all the ones not drawn on the first repetition are drawn on the second rep; otherwise X=2. So

$Pr(X=3)=\frac{k}{n}\cdot \frac{k-1}{n-1}\cdots \frac{1}{k+1}=\frac{k!}{n!/k!}=\frac{(k!{)}^2}{n!}$

and Pr(X=2) equals 1 minus that. Hence

$E(X)=2\cdot (1-\frac{(k!{)}^2}{n!})+3\cdot \frac{(k!{)}^2}{n!}=2+\frac{(k!{)}^2}{n!}.$

Loraof (talk) 23:56, 30 April 2016 (UTC)

These formulae helped me a lot! Thank you! 213.8.204.72 (talk) 11:43, 1 May 2016 (UTC)

Assuming the choice model in question is: I reach into a bag of n balls and choose k all at once (so, without repetition), then put all k back and do this again: the probability that you see a first repeated ball on step t+1 is

$(1-\frac{(\genfrac{}{}{0ex}{}{n-tk}{k})}{(\genfrac{}{}{0ex}{}{n}{k})})\cdot \prod _{i=0}^{t-1}\frac{(\genfrac{}{}{0ex}{}{n-ik}{k})}{(\genfrac{}{}{0ex}{}{n}{k})}$.

(We take $(\genfrac{}{}{0ex}{}{a}{b})=0$ if a < b, including if a < 0.) From the probability distribution it is routine to write down a formula for the expected value. A priori this is an infinite sum, but of course only the first n/k or so terms are nonzero. Possibly this expression simplifies in some reasonable way, but I haven't thought about it. --JBL (talk) 19:08, 3 May 2016 (UTC)

Also, the case k = 1 is an expectation form of the birthday problem and is treated here. --JBL (talk) 19:17, 3 May 2016 (UTC)