# What is a probability distribution?

I have a couple of fundamental questions about probability distributions, a term that is thrown around a lot. In my undergraduate courses, the term itself was not actually given a definition, rather we defined the PMF (and PDF); then said "this is an example of a probability distribution", and "here's another example of a probability distribution" (when referring the the common Normal Distribution, Binomial Distribution, etc.).

The 'definition' according to wikipedia is that the probability distribution "usually refers to the more complete assignment of probabilities to all measurable subsets of outcomes, not just to specific outcomes or ranges of outcomes".

1. When we say the Normal distribution (for example) is a "probability distribution", do we mean to say that the Normal distribution is a family of "probability distributions" with similar properties (shapes)?

2. Why is the probability distribution defined in this way: when it is completely characterized by the PMF (or PDF)? It seems redundant to me. Do we need a (quote unquote) probability distribution in order to completely assign probabilities to all measurable subsets of outcomes -- or could this be done with just the probability space $(\Omega,\mathcal{F},P)$?

• You are correct about the fact that the Normal Distribution is a family of "probability distributions". When we analyse some real life situation, it so happens that the probability of some events occurring in an experiment can be modelled by some specific member of this Normal Distribution. You can refer the "Probability and Statistics" chapter in "Advanced Engineering Mathematics" by Erwin Kreyszig for clarity :) – Banach Tarski Mar 9 '16 at 16:59
• To define this rigorously, you need to invoke measure theory E.g. the Dirac delta distribution also qualifies as a bona fide probability distribution, but you can't specify this as a function. – Count Iblis Mar 9 '16 at 17:00

The distribution of a random variable $X : \Omega \to \mathbb{R}$ is simply the probability measure $P\circ X^{-1}$ on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, that is the function: $$(P\circ X^{-1})(A) = P(X^{-1}(A)) =: P(X\in A)$$
If $X$ is discrete, then it is completely specified by the PMF. If $X$ is continuous, then it is completely specified by a PDF. And in any case it is completely specified by the cumulative distribution function $x\mapsto P(X\leq x)$. But conversely, the PMF, the PDF (up to "equivalence") and the CDF are determined by it. It is in a general context more useful than PMF's or PDF's, because it makes sense even for random variables, which are neither discrete nor continuous.
You can be even be more general: The distribution still makes sense, if you replace $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ by any other measurable space, even if the notion of a CDF makes no sense.
• Thanks, makes more sense. That's a nice way of looking at it, as the function $P\circ X^{-1}$. – Szmagpie Mar 10 '16 at 14:27