# Marginalizing by sampling from the joint distribution

For two random variables $$x$$ and $$y$$, if I can sample from the joint distribution $$p(x, y)$$, I can obtain samples from the marginal $$p(x)$$ by sampling from the joint distribution and ignoring the values of $$y$$. I want to make a formal argument for this. Something like:

\begin{align} \mathbb{E}_{x \sim p(x)} [f(x)] &= \int_{x \in \mathcal{X}} f(x)\,p(x)\,dx \\ &= \int_{x \in \mathcal{X}} f(x)\,\int_{y \in \mathcal{Y}} p(x, y)\,dy\,dx \\ &= \int_{x \in \mathcal{X}} \int_{y \in \mathcal{Y}} f(x)\,p(x, y)\,dy\,dx \\ &= \mathbb{E}_{x, y \sim p(x, y)} [f(x)] \end{align}

Is this a reasonable argument?

Yes, this is fine. To add some detail: for the second equality, you've used that $$\int_y p(x,y)\,dy = p(x)$$ for all $$x$$. The third is justified since $$f(x)$$ is just a constant as far as $$\int_y$$ cares. And for the last line, to move to the joint integral you might invoke Fubini-Tonelli.

• Thanks for the additional info! May 24, 2019 at 7:02