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I am trying to find the marginal distribution of the joint $X$ and $M$ in order to find the probability

$$Pr[X = 0,1,2,3]$$

I am given that $X|m \sim Pois(m)$ and $M \sim \Gamma (2,1) $ so I am thinking that

$$p_{X|m}(x|m)*f_M(m)$$ is the joint pdf but I realized that one is discrete and the other is continuous.

Is there something I am doing wrong?

How do I proceed from here?

Intuitively I want to find the sum of the Gamma distribution with the Poisson in it, but do not think that works.

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Looks almost fine to me. Note that it is indeed continuous as function of $m$ but it is not continuous as function of $x$. However, do note that what you calculated here is the joint pdf of $X$ and $M$. What you should want is the density of only $x$. As you usually would approach, this is done by integrating out $m$, i.e., $$P(X=x) = \int_{0}^{\infty} P(X=x|M=m)\cdot f_M(m)\,\mathrm{d} m. $$ I think this will answer your question. I leave the calculations up to you. However, if you get stuck again, let me know in the comments where.

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