Precise definition of the support of a random variable

Question

$\newcommand{\F}{\mathcal{F}} \newcommand{\powset}[1]{\mathcal{P}(#1)}$ I am reading lecture notes which contradict my understanding of random variables. Suppose we have a probability space $(\Omega, \mathcal{F}, Pr)$, where

• $\Omega$ is the set of outcomes

• $\F \subseteq \powset{\Omega}$ is the collection of events, a $\sigma$-algebra

• $\Pr:\Omega\to[0,1]$ is the mapping outcomes to their probabilities.

If we take the standard definition of a random variable $X$, it is actually a function from the sample space to real values, i.e. $X:\Omega \to \mathbb{R}$.

What now confuses me is the precise definition of the term support.

According to Wikipedia:

the support of a function is the set of points where the function is not zero valued.

Now, applying this definition to our random variable $X$, these lectures notes say:

Random Variables – A random variable is a real valued function defined on the sample space of an experiment. Associated with each random variable is a probability density function (pdf) for the random variable. The sample space is also called the support of a random variable.

I am not entirely convinced with the line the sample space is also callled the support of a random variable.

Why would $\Omega$ be the support of $X$? What if the random variable $X$ so happened to map some element $\omega \in \Omega$ to the real number $0$, then that element would not be in the support?

What is even more confusing is, when we talk about support, do we mean that of $X$ or that of the distribution function $\Pr$?

This answer says that:

It is more accurate to speak of the support of the distribution than that of the support of the random variable.

Do we interpret the support to be

• the set of outcomes in $\Omega$ which have a non-zero probability,
• the set of values that $X$ can take with non-zero probability?

I think being precise is important, although my literature does not seem very rigorous.

Answer 1

I am not entirely convinced with the line the sample space is also called the support of a random variable

That looks quite wrong to me.

What is even more confusing is, when we talk about support, do we mean that of $X$ or that of the distribution function $Pr$?

In rather informal terms, the "support" of a random variable $X$ is defined as the support (in the function sense) of the density function $f_X(x)$.

I say, in rather informal terms, because the density function is a quite intuitive and practical concept for dealing with probabilities, but no so much when speaking of probability in general and formal terms. For one thing, it's not a proper function for "discrete distributions" (again, a practical but loose concept).

In more formal/strict terms, the comment of Stefan fits the bill.

Do we interpret the support to be

- the set of outcomes in Ω which have a non-zero probability,
- the set of values that X can take with non-zero probability?

Neither, actually. Consider a random variable that has a uniform density in $[0,1]$, with $\Omega = \mathbb{R}$. Then the support is the full interval $[0,1]$ - which is a subset of $\Omega$. But, then, of course, say $x=1/2$ belongs to the support. But the probability that $X$ takes this value is zero.

Answer 2

A random variable is defined as a function that maps outcomes to numerical quantities, typically real numbers, i.e., X: Ω ↦ R.

The “domain of a random variable” is the set of possible outcomes. In the case of the coin, there are only two possible outcomes, namely heads or tails. Since one of these outcomes must occur, either the event that the coin lands heads or the event that the coin lands tails must have non-zero probability. For an unbiased coin p(H)=p(T)=1/2.

Consider another random variable Y, be the number of heads. So toss a coin twice produce y = 0, 1 and 2 with p(Y) = ¼, ½ and ¼ respectively.

The “support of a real-valued function f ” is the subset of the domain containing those elements which are not mapped to zero. Let f(x) be the p.d.f. of normal distribution with support x∈R , so that f(x)≠0.

Answer 3

You appear to have misquoted these lecture notes. What they actually say is

Random Variables – A random variable is a process, which when followed, will result in a numeric output. The set of possible outputs is called the support, or sample space, of the random variable. Associated with each random variable is a probability density function (pdf) for the random variable.

It appears to me that the author of the notes is simply using a very old notion of random variable that predates Kolmogorov's definition of it as being a function on the set of outcomes of a probability space, as has now been almost universally adopted by modern mathematicians. The notes appear to have been written for the probability unit of a course in actuarial science. They also seem to me to depart from standard modern mathematical terminology in at least one other respect—namely, in using the term "probability density function" not only in its usual sense, but also to refer to the probability mass function of a discrete distribution. Wikipedia does say, however, that the probability mass function is sometimes also referred to as a "discrete density function" (a term I can't recall ever having previously come across), so perhaps this apparent departure from standard terminology is not quite as egregious as I had originally thought.

What these notes are calling the "support, or sample space" of a random variable is not, as your misquotation has characterised it, the domain of a function constituting the random variable, but the set of possible numerical values which the random variable can have. It is essentially the same thing as what would now be called the support of the random variable's distribution in standard modern mathematical terminology.

If the set of outcomes $\ \Omega\ $ of a probability space is a topological space, with topology $\ \mathscr{T}\ $, say, and the probability $\ \text{Pr}\ $ is a Borel probability measure, then the support $\ supp_{\,\text{Pr}}\ $ of $\ \text{Pr}\ $, as Stefan Hansen and Did have pointed out in the comments, is the closed set equivalently defined as either $$ supp_{\,\text{Pr}}=\left\{x\in\Omega\,|\,\text{Pr}(\mathscr{O})>0\ \text{ for all }\ \mathscr{O}\in\mathscr{T}\ \text{ with }\ x\in\mathscr{O}\right\}\ , $$ or $$ supp_{\,\text{Pr}}=\bigcap\left\{\mathscr{C}\,|\,\text{Pr}(\mathscr{C})=1\ \text{ and }\ \mathscr{C}^{\,c}\in\mathscr{T}\right\}\ . $$ It is easy to show that these definitions are equivalent, and that $\ supp_{\,\text{Pr}}\ $ is closed. While it is not necessarily true that $\ \text{Pr}\big(supp_{\,\text{Pr}}\big)=1\ $ in the most general case, it is true in most cases of practical interest. It is true, for instance, if $\ \Omega\ $ is a $\sigma$-compact metric space.

If $\ X:\Omega\rightarrow\mathbb{R}\ $ is a real-valued random variable, then the distribution $\ \mu_X\ $ of $\ X\ $ is the probability measure $ \mu_X:\left\{S\subseteq\mathbb{R}\,|\,X^{-1}(S)\in\mathcal{F}\,\right\}\rightarrow$$[0,1]\ $ defined by $$ \mu_X(S)=\text{Pr}\left(X^{-1}(S)\right)\ . $$ If $\ X\ $ is Borel measurable, then $\ supp_{\,\mu_X}\ $ is a closed subset of $\ \mathbb{R}\ $ with $\ \mu_X\big(supp_{\,\mu_X}\big)=1\ $, and it is the smallest such closed subset. If $\ X\ $ has a probability density function $\ f\ $, then $\ supp_{\,\mu_X}\ $ will also be the support of $\ f\ $, as that term is normally defined—namely the closure of the set of points $\ x\ $ for which $\ f(x)\ne0\ $. Although the lecture notes you cite don't define random variables as functions on sample spaces, their concept of the "support of a random variable" is essentially the same thing as the set $\ supp_{\mu_X}\ $ I've just described.

I also have my doubts about the extent to which the general definition for the support of a function which you quote from Wikipedia is used in practice. I've previously only ever seen this notion defined as it is in Wikipedia's second sentence:

If the domain of $\ f\ $ is a topological space, the support of $\ f\ $ is ... defined as the smallest closed set containing all points not mapped to zero.

where I have replaced the possibly redundant "instead" with ellipses. Here, for instance, is the online Encyclopedia of Mathematics's definition, here is Wolfram MathWorld's definition, and here is planetmath's definition. As it happens, the support of a function with respect to the discrete topology is exactly the set of points on which the function does not vanish, so Wikipedia's general definition is equivalent to taking the discrete topology as being the default when no other topology has been specified. However, I've never come across an instance in technical mathematical literature where the support of a function has been defined in this way.

As far as I can recall, I have never seen an instance in modern mathematical literature where the support of any probability measure, even when it is the distribution of a random variable, has been called the support of a random variable, and, in my opinion, it would be a poor choice of terminology to use the term in that sense. It's quite conceivable to me, however, that if the sample space on which a random variable $\ X\ $ is defined happens to be a topological space, then studying the support of $\ X\ $, considered as a function on that sample space, in the sense in which the term "support of a function" is defined by the online Encyclopedia of Mathematics, Wolfram MathWorld and planet math, is something one may well wish to do. It seems to me, therefore, that use of the term "support of $\ X\ $" in that sense would be completely unobjectionable.