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My question seems similar but not exactly explained by negative binomial If I stop experiment when either k successes is reached or after N experiment What is the expected ratio success/required number of experiments Seems like limited case of negative binomial, discarding the tail

Thank you

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So I take it that if $Y$ is the number of successes divided by the number of trials, you want $E(Y)$, Certainly one can write an expression for $E(Y)$. A "closed form" may not be achievable. –  André Nicolas Mar 15 '12 at 6:24
    
that's what I need. I tried to figure out E(Y) but it started to get very complicated. It seems to be a common problem so I hoped there is some kind of standard distribution that I missed out. –  sasha Mar 15 '12 at 6:51
    
The difficulty, as you noticed, is that we let $n$ be the number of trials until the $k$-th success, in calculating $E(Y)$ we get an $n$ in the denominator. –  André Nicolas Mar 15 '12 at 6:57
    
how about estimating expected number of successes and expected number of trials (not their ratio)? It's not the same thing but at least I'd like to get the idea. Is there a simple approach for this? –  sasha Mar 15 '12 at 7:31

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