There are several probability metrics such as KL Divergence, and Hellinger distance that can be calculated in a closed form equation when two distributions are multivariate normal. With those metrics, some useful bounds for TV distance are already known.
However, I want to calculate the metric(specifically TV distance) directly in a countable space(discrete) with its definition if it is possible.
First, I'm trying to think with not that complicated examples which includes two random vector have +1 and -1 in each elements with probability 1/2.
If there are some empirical calculation tutorials or related references of Total variation distance or other probability metrics, please let me know.