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Let $X_1, X_2,...$ be a sequence of independent exponential random variables, each with mean 1. Given a positive real number $k$, let $k$ be defined by $N=\min\left\{ n: \sum_{i=1}^n X_i >k \right\}$. That is, $N$ is the smallest number for which the sum of the first $N$ of the $X_i$ is larger than $k$. I want to compute $E[N]$. I attempt to apply Wald's equation $E[\sum_{i=1}^N X_i]=E[N]E[X]$, but I have no idea to obtain $E[\sum_{i=1}^N X_i]$.

zhenyuboy
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1 Answers1

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One way to do this is to identify the distribution of $N$.

Denote $S_n=\sum\limits_{i=1}^n X_i$ as the $n$-th partial sum.

Then for $n \in \mathbb N$,

$$P(N>n)=P(S_1\le k,S_2\le k,\ldots,S_n\le k)=P(S_n\le k)$$

Now $S_n$ has a Gamma distribution since it is the sum of independent Exponential variables. And from the Gamma-Poisson relationship, it follows that

$$N-1 \sim \text{Poisson}(k)$$

This would also be apparent if you notice the connection to a Poisson process:

$$N-1=\max\{n:S_n\le k\}$$

StubbornAtom
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