19.4 Density functions

Distribution functions are not our only tool to describe real-valued random variables. If you have studied probability theory from a book/lecture/course written by a non-mathematician, you have probably seen a function such as

referred to as “probability” at some point. Let me tell you, this is definitely not a probability. I have seen this mistake so much that I decided to write short X/Twitter threads properly explaining probabilistic concepts, from which this book was grown out of. So, I take this issue to heart.

Here is the problem with cumulative distribution functions: they represent global information about local objects. Let’s unpack this idea. If X is a real-valued random variable, the CDF

describes the probability of X being smaller than a given x. But what if we are interested in what happens around x? Say, in the case of the uniform distribution (19.5), we have

(We used Theorem 115 when taking the limit.)

Thus, as...