Is "probability distribution function" a distribution?

Question

I can understand the definition of distribution as written in https://en.wikipedia.org/wiki/Distribution_(mathematics) On the other hand there are three different terms in the definition of probability distribution function(PDF) : https://en.wikipedia.org/wiki/Probability_distribution_function My question: is PDF a distribution? If so can anyone help me to clarify how a PDF is a distribution?

Answer 1

The only thing that relates them are that they are both restricted cases of measures (or at least the PDF can be interpreted as such).

A distribution in probability theory is very like a measure with the restriction that the measure of the whole space is 1 (ie $\int dp = $int p(x) dx = 1$).

The other distribution is quite restricted since it's only allowed to act on smooth functions with compact support (ie $\int \varphi d\mu$ need only bee defined if $\varphi$ is smooth with compact support).

But since a probability distribution is that general and smooth functions with compact support is so well behaved you can always integrate a such (ie $\int \varphi dp$ is well defined as required of the second type of distribution). You could of course generalize the concept of PDF to allow for any measure (that is not necessarily representable as a function).

Answer 2

The term distribution in probability theory is not related to the concept of distribution as functional (as cited in your first hyperlink). Formally, this refers to a probability measure on a state space, which is usually $\Bbb{R}$ or $\Bbb{R}^d$. Moreover, if a random variable $X$ is given on a probability space $(\Omega, \mathcal{F}, \Bbb{P})$, then the corresponding pushforward measure $\Bbb{P}(\cdot \in X)$ is called the distribution of $X$. This allows us to compare the random behavior of random viariables which live on different spaces.

There are may quantities that uniquely specify a probability measure. For instance,

• For each PDF $f$ there corresponds a measure $\mu$ given by $\mu(E) = \int_E f$.
• For each PMF $p$ there corresponds a measure $\mu = \sum_x p_x\delta_x$.
• For each CDF $F$ there corresponds a measure $\mu$ that satisfies $\mu((-\infty, x]) = F(x)$.

Consequently some authors tend to use the term 'distribution' to refer any of them.