I am working with a process that can be modeled as sparsely occurring jumps of either +1 or -1.
I am interested in finding the probability distribution of the net (integrated) effect of this random process over an arbitrary time interval $T$. I assume that I have an overall occurrence rate $\lambda$ as well as the probabilities $p(+1)$ and $p(-1)=1-p(+1)$ of a given event being $\pm 1$.
I assume by CLT that this process should be related to the binomial distribution over time, but I'm not sure how to calculate the variance. If I make the assumption that occurrence rates can be separated into $\lambda_+$ and $\lambda_-$ for the two event outcomes, then I believe I can calculate the mean as:
$\mu(T) = (\lambda_+ - \lambda_-) T$
However, I don't understand how to get the variance of this distribution. (Does this somehow relate back to Poisson process?). Any insight would be much appreciated!