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I understand why the expected number of trials until there is a success is given by

$$ \sum_{i=0}^{\infty} i p q^{i-1} \ = \ E[\text{number of trials until} \ X=1] = \frac{1}{p} $$

where $p$ is the probability of success and $X=1$ denotes the first success. However, I have different problem. In my setting, after the $n$th trial there will be success with probability 1. The probability of a success happening at the $i$th period [before n periods] is $p(1-p)^{i-1}$. I suspect that the expected amount of trials until success is given by

$$ p + 2p(1-p) + 3p(1-p)^2 + ...+ np(1-p)^{n-1} $$

I am not sure if the correct answer is either

$$ \frac{1-(1-p)^n}{p} $$

or

$$ \frac{1-(1-p)^n}{p} -n(1-p)^n $$

I have tried using a few partial sum tricks and forcing the addition of probabilities equal to one in the following way:

$$ \sum_{i=0}^{n} p (1-p)^{i-1} + (1-p)^{n-1} \ = 1 $$

But I am not sure if this is the right way to even approach the question. How can I calculate the expected value of trials until the first success when I know that at a some $n$th period there will be success with probability 1?

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  • $\begingroup$ You say "after the nth trial" probability is 1". But if success definitely occurs at exactly trial number n, it would seem the sample space would be the integers from 1 to n (a finite sample space). If so the presence of a 1 at trial n would force the previous trial probabilities to be each 0 in order that the sum of probabilities over the sample space be 1. $\endgroup$
    – coffeemath
    Jan 3 '16 at 4:23
  • $\begingroup$ Maybe it should be framed as a conditional probability given that success occurs on or before trial n, and then use expected value for that conditional distribution. $\endgroup$
    – coffeemath
    Jan 3 '16 at 4:25
  • $\begingroup$ How could I express it as a conditional expectation? I suspect I have to divide by something so as to correct the support of the distribution, but what would that be. Also, what would even be the pdf of such a distribution? Also, I made a few edits above. $\endgroup$
    – Sophie
    Jan 3 '16 at 4:34
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    $\begingroup$ So if there has been no success before the $n$-th trial, there is success for sure on the $n$-th? If so the last term should be $n(1-p)^{n-1}$. Apart from this term, you have a finite arithmetico-geometric series, and there is a clsed form for the sum. $\endgroup$ Jan 3 '16 at 4:46
  • $\begingroup$ Andre, do you mean that last term should go into my last expression, that one I equalized to 1? Or the one above [that one has in its first term a finite geometric series]? $\endgroup$
    – Sophie
    Jan 3 '16 at 5:16
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Define the random variable $X = \sum_{i \leqslant n}i{\bf 1}_{\{\text{1st success on trial }i\}}$ , so that the expectation is computed as follows: $$\mathbb E[X] = \mathbb E[\sum_{i \leqslant n}i{\bf 1}_{\{\text{1st success on trial }i\}}] = \sum_{i \leqslant n}\mathbb E[i{\bf 1}_{\{\text{1st success on trial }i\}}] = \sum_{i \leqslant n}i{\bf P}[{\text{1st success on trial }i}] \\ = \sum_{i < n}i(1-p)^{i-1}p + n(1-p)^{n-1} = \frac{1-(1-p)^{n-1}(1+(n-1)p)}{p} + n(1-p)^{n-1}.$$

That is,

$$ \mathbb E[X] = \frac{1-(1-p)^{n-1}(1+(n-1)p)}{p} + n(1-p)^{n-1} $$


The form of $\mathbb E[X]$ can be simplified further, since $$\mathbb E[X] = \frac{1-(1-p)^{n-1}(1+(n-1)p)}{p} + n(1-p)^{n-1} = \frac{1-(1-p)^{n-1}(1+(n-1)p - np))}{p} = \frac{1-(1-p)^n}{p}$$ Therefore,

$$ \mathbb E[X] = \frac{1-(1-p)^n}{p} $$

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  • $\begingroup$ Just to make sure, shouldn't the last term on the right hand side be divided by p? Also, Shouldn't the first term be $\frac{1-(1-p)^{n-1}(1+n(p-1))}{p}$? $\endgroup$
    – Sophie
    Jan 3 '16 at 5:54
  • $\begingroup$ @Sophie No, the last term does not need to be divided by $p$. If, however, one wishes to have a $p$ in the denominator, one may write it as $np\frac{(1-p)^{n-1}}{p}$. And no, the first term should not be as you suggested; your suggestion gives an incorrect answer when $n=1$. $\endgroup$
    – ki3i
    Jan 3 '16 at 13:24

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