Given the sequence $(X_n), n=1,2,… $, of iid exponential random variables with parameter $1$, define: $$ M_n := \max \left\{ X_1, \frac{X_1+X_2}{2}, …,\frac{X_1+\dots+X_n}{n} \right\} $$ I want to calculate $\mathbb{E}(M_n)$. Running a simulation leads me to believe that $$ \mathbb{E}(M_n)=1+\frac{1}{2^2}+\cdots+\frac{1}{n^2} = H_n^{(2)}.$$ Is this correct? If yes, how would one go proving it? I tried […]

How does one prove that the unique continuous distribution with the memoryless property is the exponential distribution? i.e. Suppose we know that a continuous random variable $X$ satisfies $$P\{X > t+s \mid X > s \} = P\{X > t \}$$ Then how do we show that $X \sim \text{Exponential}(\lambda)$ for some $\lambda > 0$? […]

Question: Suppose, there is a queue A having i customers initially. The service time of the queue is Exponentially distributed with parameter $\mu$ (i.e., mean service time of one customer in Queue A is 1/$\mu$). Arrival to queue A has Poisson distribution with mean $\lambda$. What is the expected time before queue A becomes empty […]

How to prove that geometric distributions converge to an exponential distribution? To solve this, I am trying to define an indexing $n$/$m$ and to send $m$ to infinity, but I get zero, not some relevant distribution. What is the technique or approach one must use here?

We have a system in which events happen one after each other. The time interval between each two events shown by random variable $t_i$. So, the time interval between the first and the second events is shown by $t_1$, the time interval between the second and the third events is shown by $t_2$, and so […]

In answering finding Expected Value for a system with N events all having exponential distribution, I somehow missed the fact that the OP wanted the expected value conditional on the number of events. Since I put quite a bit of effort into deriving the unconditional expected value, and this might conceivably be of interest for […]

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