# Confusing Terminology: distribution or density?

This is a terminology question, which I am trying to understand.

I have noticed when the term "distribution" is used, it refers to "cumulative" quantities (CDF) and when use "density" they refer to non-cumulative, in particular the PDF. (The complete form of PDF or CDF is seldom used in discussions, and seems like mutual understanding is complete in stats community.)

If my understanding above is correct about the use of terminology, I expect this rule to generalize, and then when people use "conditional distribution", this refer to a cumulative conditional distribution.

Looks like this is not the case. Rather, it is a non-cumulative.

Can someone clarify the common usage of the terminology please?

## 1 Answer

There are two slightly different uses of the word "distribution" here.

First, and more fundamental, the "distribution" of a random variable $$X$$, or the "probability distribution" of $$X$$, or the "probability law" of $$X$$, is a probability measure, that assigns to each set $$A$$ of possible values the probability that $$X$$'s value is in $$A$$, that is, that assigns to $$A$$ the number $$P(X\in A)$$. Or more precisely, assigns to each measurable set $$A$$ the number $$P(X\in A)$$.

Second. One way to describe this distribution is with the "cumulative distribution function", given by $$F_X(x)=P(X\in(-\infty,x])$$. It is a basic result that if you know $$P(X\in A)$$ for all the sets $$A$$ of special form $$A=(-\infty,x])$$ for all real values of $$x$$, you know $$P(X\in A)$$ for all measurable sets $$A$$. That is, the "distribution" in sense 1 tells you what the "cumulative distribution function" is, and vice versa.

The term "density" is used when the distribution is given by a formula like $$P(X\in A) = \int_A f_X(u) \,du$$ or (equivalently) the distribution function is given by a formula like

$$F_X(x)=\int_{-\infty}^x f_X(u)\,du.$$ Then $$f_X$$ is the "density function" of $$X$$.

It is a basic fact that not all probability distributions "have" density functions, but for those that do, whatever tools you know from integral calculus are applicable.

Added next day. Now let's throw "conditional" into this. The simplest case is when we condition on an event $$B$$. This creates a conditional distribution on $$X$$, call it $$P_B$$, such that $$P_B (X\in A) = \frac{ P([X\in A]\cap B) }{P(B)},$$ and corresponding conditional cumulative distribution function $$F_{X|B}$$ given by $$F_{X|B}(x) = P_B (X\in (-\infty,x]) = \frac{ P([X\in (-\infty,x]]\cap B )}{P(B)},$$ and this distribution function might be given by a conditional density function $$f_{X|B}$$ such that $$F_{X|B}(x)=\int_{(-\infty,x]]}f_{X|B}(u)\,du$$ and so on.

More complicated is when we condition on a random variable, $$Y$$, say, resulting in the conditional distribution of $$X$$ given $$Y$$, which might be written $$P_Y$$ or $$P_{X|Y}$$, its cumulative distribution function $$F_{X|Y}$$, which might have a conditional density function $$f_{X|Y}$$, and so on. Beware, however, that there are definitional problems with conditioning on a continuous random variable, some of which is explained in the Wikipedia article.

• So as a "function" density is f above and distribution is F above. Now conditional distribution is f(.|.) again? – user25004 Apr 12 at 15:09