Q Random variables X and N have joint distribution, defined up to a constant of proportionality,

$$f(x,n) \propto \frac{e^{-3x} x^n}{n!} ~,\quad n=0,1,2, \ldots , x>0$$

Implement a Gibbs sampler to sample from this distribution.

I understand in order to implement Gibbs sampler is requires being able to simulate the conditional distributions of $X$ and $N$ given that the other is fixed.

So far I have:

$X$ is a continuous random variable and $N$ is a discrete random variable. The conditional distribution of $X$ given that $N = n$ is proportional to $e^{-3x} x^n$ for $x > 0$

I tried seeing with if Gamma distribution could be simulated for X and with parameters ${r=n+1}$ and $\lambda = 3$ and I obtained that the conditional distribution would be equal to

$$\frac{ 3^{n+1} e^{-3x} x^n }{ n! }$$

which actually looks like the original proportion with an added $3^{n+1}$.

The conditional distribution of $N$ given that $X = n$ is proportional to $$\frac{ x^n }{ n! } \quad \text{for} \quad n=0,1,2,\ldots$$

This looks like a Poisson distribution with parameter $\lambda = x$ but then we would be missing the $e^{-x}$ that would appear in this Poisson distribution.

I understand how to complete this in R once I am able to simulate these conditional distributions but I'm stuck on that part. Any insight would be appreciated.

  • $\begingroup$ By that phrase, I just wanted to show what I've done with the question so far which is what is listed below the phrase. The question itself is what I listed above the horizontal line which includes everything that is known about the distribution. If it helps, this question is intended to be done in R but from my understanding I needed to figure out how to simulate the conditional distributions for X and N before coding in R $\endgroup$ – Niko L Mar 24 at 17:56

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