# ETOOBUSY ðŸš€ minimal blogging for the impatient

# Same rejection method?

**TL;DR**

Where we ask ourselves: did we talk about the same

rejection methodslately?

In A 4-faces die from a 6-faces die we introduced a rejection method to generate a D4 die from a D6. Then in Rejection method we took a look at a generalization of that method, to generate samples from an arbitrary (albeit limited in the $x$ axis) probability density.

Are the two really related, though?

# The discrete case

First, letâ€™s convince ourselves that the rejection method discussed in Rejection method works equally well for discrete probability densities. As we saw, there are two random draws for each sample:

- one over the $x$ axis
- another one over the $y$ axis.

For discrete densities, instead of using a *uniform distribution* for
the first draw, it suffices to use the equivalent discrete distribution
to draw one of the possible discrete alternatives. After that, the
procedure on the $y$ axis remains unchanged.

# Soâ€¦ D4 from a D6?

Letâ€™s consider a D4 like a D6, where two of the outcomes (namely `5`

and
`6`

) have probability to come out equal to $0$.

The *enclosing* function would in this case be one with value $0.25$
over all six values (in brown below), which corresponds, with proper
scaling, to the discrete uniform drawing of our D6:

At this point, we can easily see that we donâ€™t *really* need any random
draw over the $y$ dimension, because it will always lead to *acceptance*
when the first draw is one of `1`

, `2`

, `3`

, or `4`

, and it will always
lead to a *rejection* for `5`

and `6`

. So we can avoid it and fall back
to the method described in A 4-faces die from a 6-faces die. Yay!