Consider two stochastic processes: $T_t$ is the number of times we flip a coin with bias $\rho$ by time $t$, and $X_t$ is the number of heads of the coin by time $t$ (i.e. after $T_t$ flips):
$$T_t \sim \text{Poisson}(\lambda) $$
$$X_t \sim \text{Binomial}(\rho, T_t)$$
How would one go about computing the covariance/joint distribution of $(T_t, X_t)$?
I know the pdf of $X_t$ and the pdf of $T_t$ for any fixed $t>0$. These are simply the binomial and poisson distributions, $$P(X_t = k) = {T_t \choose k} \rho^k (1-\rho)^{T_t - k}$$ $$P(T_t = n) = \frac{(\lambda t)^n}{n!} e^{-\lambda t}.$$ But I do not know how to proceed from here.
