Expectation value of the reciprocal of a sum of geometrical

Asked May 19 '20 at 22:12

Active May 19 '20 at 22:12

Viewed 76 times

I was studying statistical inference when I had a problem with the following probability problem.

Problem:

Suppose I have that $X_i \sim Geom(p)$ where $ p \in [0,1]$.

Let us define $Y= \frac{1}{\sum_{1 \leq i \leq n} Xi}$.

I would like to find the value of $\mathbb{E}[Y]$.

My attempt:

I tried using some manipulation and using the Beta function but I cannot solve anything. I tried computing explicitly the sum involved but it is not easy. I used How to compute the sum of random variables of geometric distribution to compute the distribution of the sum of Geometrical a.v.

asked May 19 '20 at 22:12

Filippo Giovagnini

2,735

There is a positive probability that the sum of the $X_i$ equals $0$. – JimB May 20 '20 at 00:20
Do you want the mean conditional on the sum of the $X_i$ being greater than $0$? – JimB May 21 '20 at 22:33

Expectation value of the reciprocal of a sum of geometrical

Problem:

My attempt:

0 Answers0