I have been struggling with some notation and cannot figure out where to find an in-depth explanation of how to approach this problem. Does anyone know any references or specific topics I need to study to really get a handle on this?
Just to be clear, this is homework, but I am not sure what exactly the notation means/ what I should be doing to solve it.
$X_r$ is a Poisson random variable with intensity rate $\frac{1}{r^2}$. I need to figure out $E[\sum_{r=1}^{\infty}{rX_r}]$. My teacher's "hint" is to use the Monotone Convergence Theorem, but that theorem is written as follows: $\lim_{n\rightarrow \infty} \int{f_n} d\mu = \int{\lim_{n \rightarrow \infty f_n} d\mu}$.
I have several questions... First, it seems this $\sum_{r=1}^{\infty}{rX_r}$ should converge to a Poisson R.V. But how do I determine what it converges to? When calculating expected value we need to use the probability measure function, but here we have many different ones (one for each $rX_r$, right?). Am I supposed to figure out the what the $rX_r$ converge to and use that Poisson's probability function to find the expected value? (right side of the monotone convergence theorem).
Just for clarity the integrals in the chapter are all Lebesgue integrals, and this is my first exposure to them.
I am just very confused and do not know how to understand this. I thought it would be like so: $E[\sum_{r=1}^{\infty}{rX_r}] = \sum_{r=1}^{\infty}{E[rX_r]} = \sum_{r=1}^{\infty}{rE[X_r]} = \sum_{r=1}^{\infty}{r \frac{1}{r^2}} = \sum_{r=1}^{\infty}{\frac{1}{r}} = \infty $.
But this diverges, so I do not think this is correct. I am rather confused. Is the notation my Professor used confusing, or am I just missing something fundamental? Please give me advice and point me in the right direction. Thank you so much!!