Poission processes and standard deviation
first, this is my first post here. So I apologize in advance for all wrongdoings.
Here is my problem:
I have a random process that follows a Poisson distribution with the parameter
, hence if is my random variable
then the standard deviation of that random variable should converge to . I.e.
and, since it is a Poission process
for large N as well.
But then of course
Now, assume you have not one Poission process but M processes. Each with it's own . And that the random variable now is a single realisation of each of those processes.
Is it true (and I have strong reasons to believe it should be, at least under certain circumstances), that
Is there prove?