Say $X_0, X_1$ and $X_2$ are independent Poisson random variables with means $\lambda_0, \lambda_1$ and $\lambda_2$, respectively. Define $Y_1 = X_0+X_1$ and $Y_2 = X_0 + X_2$.
How would I find an expression for the joint probability of $Y_1$ and $Y_2$, that is, $P(Y_1=y_1, Y_2=y_2)$?
Approaches I have attempted, unsuccessfully:
1) $P(Y_1=y_1, Y_2=y_2) = P(X_0+X_1=y_1)*P(X_0+X_2=y_2|X_0+X_1=y_1)$ but I am unable to solve the second probability.
2) Trying to set up a multivariate transformation with $Y_1 = X_0+X_1$ and $Y_2 = X_0 + X_2$, but I get stuck when trying to write $X_0, X_1$ and $X_2$ in terms of $Y_1$ and $Y_2$ to get the Jacobian.
