Here is an exercise given by a colleague to a student :

Let $X\hookrightarrow B(n,p)$ and $Y=\frac{1}{X+1}$. Find ${\rm E}(Y)$.

It is not very difficult to prove that the answer is $${\rm E}(Y) = \frac{1-q^{n+1}}{p(n+1)}$$ where $q=1-p$. But the answer can also be written $${\rm E}(Y) = \frac{1+q+q^2+\dots+q^n}{n+1}$$

First question: Is there any meaning to this form, which looks very much like a mean value of some sort? Or maybe another proof of this result which explains it in a more direct way?

Second question : Is there some context which could make this exercise more "concrete"?

  • 3
    $\begingroup$ In other words, is there a "natural" (whatever that means) explanation of the identity $$E\left(\frac1{X+1}\right)=E(q^U)$$ for $X$ binomial $(n,p)$, $U$ uniform on $\{0,1,\ldots,n\}$, and $q=1-p$? Interesting question... $\endgroup$ – Did Jun 21 '18 at 5:55
  • $\begingroup$ One can possibly justify the interchange of integral and summation to say that $E\left(\frac{1}{1+X}\right)=E\left(\int_0^1 t^X\,dt\right)=\int_0^1 E(t^X)\,dt=\int_0^1 (1-p+pt)^n\,dt$. But that's just another proof, not sure if this might be helpful in this regard. $\endgroup$ – StubbornAtom Jun 21 '18 at 11:48

Here is an application of your question to a setting that highly interests me. In queueing there is the notion of utilization which is the long-run fraction of time a server is busy serving demand. Consider a Markovian service setting where demand arrives according to Poisson with $\lambda=1$ and there are $N+1$ servers with Exponential service time and mean rate $\mu=1$, where $N\sim\text{Bin}\left(n,p\right)$.

An application of this is the utilization of an Uber driver; the number of Uber drivers on a given instance is uncertain as it cannot be mandated or enforced by the firm. Given $k$ drivers choose to drive on the streets, their utilization would be $\frac{\lambda}{\mu\cdot k}=\frac{1}{k}$. So, in this setting if we assume the above model by the law of total probability the utilization of an Uber driver would be $E\left[\frac{1}{1+N}\right]$. Given this, I would be interested to know of an interpretation of the RHS? Any ideas?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.