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A routine exercise designed to test one's knowledge; often from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for such questions rather than complete answers.
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Finding the joint distribution from poisson conditionals
Suppose that for two discrete random variables $X_1$ and $X_2$, we know their conditional distributions. Namely
$$X_1~|~X_2 = x_2 \sim \mathrm{Poisson}(\lambda_1 + ax_2),$$
$$X_2~|~X_1 = x_1 \sim \mat …