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In the simple case that $X$ is a real valued random variable, the first thing I would reach for is the inverse-cdf method, especially since you have mentioned "runif" which gives draws from a uniform …

I will expand on this answer later if there is interest and when I have some references handy. But for now you may be interested in the following way of thinking about the problem. One way to charac …

EDIT: This is wrong -- careless mistake as noted in the comments. I thought I had deleted it, but here it still is. Working with the RHS of your inequality we have \begin{eqnarray}\sum_i (P_i - …

Maybe something like this will work. Consider $U_1$ and $U_2$ drawn uniformly at random on the unit circle. Because they are uniformly distributed, we may rotate the circle until $U_1$ is at the ``n …

Total variation and Hellinger distance are two standard ways to measure this. Kullback-Leibler divergence is another standard way, as would be general $f$-divergences. The Earth-Mover's distance (al …

You can always write down the joint distribution compositionally. In terms of a density function: $$f(\theta_1, \dots, \theta_n) = f_1(\theta_1)f_2(\theta_2 \mid \theta_1)f_3(\theta_3 \mid \theta_1, \ …

Section 2.1 of this paper gives expressions for the posterior mean of location parameters. This may be helpful in your context.

In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are equivalen …