Suppose I have Poisson random variables $n_1$, $n_2$ with parameters $\lambda_1$,$\lambda_2$ and then I want the look at the random variable $X\stackrel{d}=\text{Bin}(n_1+n_2,p) - n_1$. How would I calculate the variance of such a variable? I am thinking of possibly using a form of the law of total variance to get:
$$ \text{Var}(X) = \mathbb{E}[\text{Var}(X\mid n_1,n_2)] + \mathbb{E}[\text{Var}(X \mid n_1,n_2) \mid n_1] + \text{Var}(\mathbb{E}[X\mid n_1]) $$
Is this correct? If so, how would I even calculate a conditional variance where I have two random variables, does this even make sense?
I am unsure on how I would calculate variance for a random variable where it depends itself on the random variables $n_1$ and $n_2$. I have calculated the expectation, this was straightforward.
Could I possibly split this up, using covariance, to get $\text{Var}(\text{Bin}(n_1+n_2,p)) + \text{Var}(n_1) - 2\text{Cov}(\text{Bin}(n_1+n_2,p),n_1)$?
Still the question remains to solve this covariance equation.