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Suppose you have a sum of IID random variables (uniformly distributed in [0, 1])
$$S = \sum_{i=1}^N X_i$$
if I want to have a rough idea of the average value of $N$ such that the sum is equal to some number $S_0$ is it safe to say that $N$ is going to be around $S_0/{\rm E}[X_i]$ (assuming I'm not under the conditions of using Martingale).

UPDATE: I reformulate the problem. Suppose $N$ is a random variable defined as the smallest integer such that $$ \sum_{i=1}^N X_i \geq S_0 $$ where $X_i$ are IID and uniformly distributed in $[0, x_0]$, with $x_0 < S_0$. What is $E[N]$?

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    "assuming I'm not under the conditions of using martingale" -- Can you clarify what this means?2012-01-04
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    @Bob $S$ is continuous random variable that follows [Irwin-Hall distribution](http://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution). The probability that it is equal to any specific value is zero, as for any other continuous RV. You should think in terms of $\mathbb{P}(S > S_0) > \alpha$ instead.2012-01-04
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    In the light of @Sasha's comment, see if the following is a suitable formalisation of your question: If the random variable $N$ is defined as the smallest number such that $\sum \limits_{i=1}^{N} X_i$ exceeds $S_0$, then what is $\mathbf E[N]$? Is it approximately equal to $\frac{S_0}{\mathbf EX_i}$.2012-01-04
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    If $N:=\inf\{n:S_n\geq S_0\}$, then [Wald's Equation](http://en.wikipedia.org/wiki/Wald%27s_equation) gives that $\mathbb{E}[S_N]=\mathbb{E}[N]\mathbb{E}[X_1]$, giving bounds on $\mathbb{E}[N]$.2012-01-04
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    yes, you are both right. I wanted to know $E[N]$ according to Srivatsan's formulation of the problem and, if possible, some higher order statistics.2012-01-04

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