[Math] Random Variable Vs. Probability Function Intuition

probability

In a book on probability I'm reading they begin by defining random experiment, outcome, sample space & event, then using these notions they define & a probability space in terms of the sample space, a sigma field & a probability function.

After about 80 pages of working with this theory, i.e. Bayes theorem, conditional probability etc… they introduce the notion of a random variable & the distribution function of a random variable.

They give the impression that a random variable is a function of the outcomes,
& that it is defined independently of the notion of a probability function, which this stack answer seems to imply:

Probability measures assign values (probabilities) to sets in the $\sigma$-algebra $\mathcal{A}$. On the other hand, random variables are functions $f\colon \Omega\to E$ that are measurable in this sense: If $B \in \mathcal{E}$, then $f^{-1}(B) \in \mathcal{A}$. https://math.stackexchange.com/a/124504

yet other sources I've seen seem to conflate the two notions or even imply they
mean the same thing 🙁

What is a random variable a) intuitively, b) mathematically & c) in a way that distinguishes it from a probability function, i.e. a nice intuitive example of a situation where you can't just use a probability function you need a random variable. I can't believe I got through a course on probability without understanding this 🙁

Thanks!

Best Answer

The word experiment because of its connection in chemistry, or physics clouds our understanding. It need not be that in statistics we "conduct" the experiment as in science tightly controlling the settings. No.

To understand where probability models can fit in, one should come out of textbook examples of probability associated with gambling (cards, dice, coin tosses). Then we see natural examples of random variables that will give us intuition. Many times we are not interested in the outcome directly, but on some aspect (that could be measured with numbers) of the outcome.

For example a telemarketing executive might call all numbers from a printed phone book in the order. Making the call is a random experiment. We don't know who would pick it up, whether it would be busy etc all these are not predictable in advance.

One random variable could be the duration of the call. (a numerical measurement on the outcome of the dialling of the call.) Another random variable (in the same experiment, that is same sample space) could be how much of sales (measured in dollars) the executive could accomplish by that call.

Another "experiment". Assume a restaurant opens at noon everyday. Who the first customer is unpredictable, so it is a random experiment. The sample space here is potentially whole population of the city. A random variable for this experiment could be "how much the customer dined" (the bill paid). Another could be for how long the customer stayed in the restaurant. Another random variable could be how many persons came in a group as the first batch of customers (a family dining out, a person treating friends, a group of co-workers coming out together from office for lunch)

Related Solutions

Probability – Why is the Probability of a Continuous Random Variable Taking a Specific Value Zero?

The problem begins with your use of the formula

$$ Pr(X = x) = \frac{\text{# favorable outcomes}}{\text{# possible outcomes}}\;. $$

This is the principle of indifference. It is often a good way to obtain probabilities in concrete situations, but it is not an axiom of probability, and probability distributions can take many other forms. A probability distribution that satisfies the principle of indifference is a uniform distribution; any outcome is equally likely. You are right that there is no uniform distribution over a countably infinite set. There are, however, non-uniform distributions over countably infinite sets, for instance the distribution $p(n)=6/(n\pi)^2$ over $\mathbb N$.

For uncountable sets, on the other hand, there cannot be any distribution, uniform or not, that assigns non-zero probability to uncountably many elements. This can be shown as follows:

Consider all elements whose probability lies in $(1/(n+1),1/n]$ for $n\in\mathbb N$. The union of all these intervals is $(0,1]$. If there were finitely many such elements for each $n\in\mathbb N$, then we could enumerate all the elements by first enumerating the ones for $n=1$, then for $n=2$ and so on. Thus, since we can't enumerate the uncountably many elements, there must be an infinite (in fact uncountably infinite) number of elements in at least one of these classes. But then by countable additivity their probabilities would sum up to more than $1$, which is impossible. Thus there cannot be such a probability distribution.

Probability Theory – Precise Definition of the Support of a Random Variable

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.

Best Answer

Related Solutions

Probability – Why is the Probability of a Continuous Random Variable Taking a Specific Value Zero?

Probability Theory – Precise Definition of the Support of a Random Variable

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