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I've been stuck on this question for a while.

{$U_{i}$ iid, Uniform(0,1)
$V_{n} = \prod_{1}^{n} U_i$, n = 1, 2,...
$N = \min \{k: V_k < 0.1\} $
Find the distribution of $N$

I know that this is supposed to be related somehow to a poisson distribution based off of other things that I have learnt, but that was with the summation of uniform distributions, not the product. I honestly just don't know where to begin. Any help would be very much appreciated!

gt6989b
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djl333
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1 Answers1

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Rewrite $N$ as $N=\min\{k:S_k>c\}$ where $S_k=-\ln V_k=-\sum\limits_{i=1}^k \ln U_i$ and $c=-\ln(0.1)$.

Then for $k\in \mathbb N$, $$\color{blue}{P(N-1\ge k)}=P(N>k)=P(S_1\le c,S_2\le c,\ldots,S_k\le c)=\color{green}{P(S_k\le c)}$$

Now $S_k$ is the sum of i.i.d Exponential$(1)$ variables $-\ln U_i$, so that $S_k$ has a certain Gamma distribution. Finally recall the Gamma-Poisson relationship.

StubbornAtom
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