# Can we think of a Random Variable as an instantiation of its distribution?

I am going through Harvard's Statistics 110 course.

In lecture 11 (https://youtu.be/TD1N4hxqMzY?t=4m38s), professor Blitzstein says that many students confuse random variables (RVs) with their distribution. As an analogy to help students separate these concepts, he says that RVs can be thought as houses, and distributions as blueprints for houses.

Does this mean that we can view RVs as a concrete value from the distributions (for example, after having done an experiment, we now have instances and no probabilities and involved anymore), or am I misinterpreting his analogy?

• See stats.stackexchange.com/questions/50 for answers to the question "what is a random variable?" It's hard to tell what you might mean by "concrete value from the distributions." – whuber Jul 17 '17 at 16:18
• Not sure what I meant myself. I was just trying to understand his analogy of (RV = house, dist = blueprint) by mapping the formal definition of RV (function from outcome space to R or some other number system) to his analogy, and I got confused. – samlaf Jul 17 '17 at 16:26
• Your comment in the link helped me: 'When the "events" become "known," what happens to the random variable? According to this answer, it can no longer exist!' So then a RV can't really be a "house", since there are no "random events" associated with a house, everything is determined (whether you used wood instead of stone, etc.). Right? – samlaf Jul 17 '17 at 16:29
• The "random variable" is a fixed mathematical object. I have adopted a metaphor of Freeman et al in writing about it as a "consistent way of labeling tickets in a box." The realization, in this metaphor, is the physical process of drawing a ticket from that box (and then returning it, so that the contents of the box remain unchanged). That should make it clear that "realizing" a random variable does not change the random variable. It also makes a clear distinction between a realization and the distribution (which describes the frequencies of numbers on those tickets). – whuber Jul 17 '17 at 17:30

## 2 Answers

Yes it is a value, but no it doesn't necessarily have to be realized. A random variable can be realized or unrealized. Just as a house can be built or unfinished. The analogy is meant to emphasize that a random variable can be thought of as the value, while a distribution is a function that describe the probability of those values. A random variable is not the thing doing the generating (blueprint, probability distribution); rather it is the thing being generated (house, random variable).

You can take this a step further. A random variable can be "looked at" in a few ways. All of these entities are separate things but "describe" the same phenomenon. Depending on the question you want to answer, you might use a random variable's

1. value/label/representation, usually denoted by capital letters at the end of the alphabet. This is what he means when he talks about a random variable. This describes the outcome of one draw. Even though this convention is not always followed, usually it is capitalized if it has not be observed concretely. And it is written with a lower-case letter if it has.
2. probability density/mass function. This is usually what is meant by a random variable's "distribution." A random variable will have one of these if it is discrete (pmf) or continuous (pdf). Sometimes it is denoted by $f_X(x; \theta)$ or $p_X(x;\theta)$, or something similar. They are useful for finding a random variables expected value, or variance, or other expectations. They can also be summed (discrete rvs) or integrated (continuous rvs) to give you probabilities of certain events or outcomes of the random variable.
3. cumulative distribution function. This is a function that gives you probabilities that a random variable can be in a certain range.
4. moment generating function, when they exist they completely define a random variable," good for finding the distribution of linear combinations of independent random variables. They are also another way to find a random variable's moments.
5. characteristic function, similar to the mgf above.
• This answer furthers the confusion. Random variables and distributions are quite distinct things, both mathematically and conceptually, yet you seem to equate them. – whuber Jul 17 '17 at 16:17
• @whuber true. See new edits. – Taylor Jul 17 '17 at 16:41
• But according to whuber, a random variable doesn't exist once it is realized (it is no longer random). So the way I see it now, a RV would be analogically equivalent to a prospective house (have decided to build a house at some location, but still not set on the exact specifications, which are given by the distribution). Once the house is completed however, we can no longer talk of it being a random variable. Is this correct? – samlaf Jul 17 '17 at 17:06
• @samlaf yes, that's right. We could call that a realization of a random variable, though. The words aren't off limits, but it's not random anymore. The distinction being made by the analogy is not the random/nonrandom one, but rather the generator/generated one. – Taylor Jul 17 '17 at 17:20
• Thank you very much for the added remarks in your post: in particular, they help me understand the original house/blueprint analogy (+1). – whuber Jul 17 '17 at 17:25

One intuitive distribution is the Bernoulli distribution. It describes the outcome of throwing a coin, which lands head with probability $p$ and tails with prob. $q=1-p$.

• If you throw the coin once, you will observe either head or tail. However, this outcome is the random variable, it is not the distribution. The distribution however defines, with which probability you observe head and tails. The same true for all distributions -- continuous and discrete.

Blitzstein's analogy goes a bit further, because there exists not a single Bernoulli distribution, but a family of Bernoulli distributions: For each value of $p$ you will get a different Bernoulli distribution.

• I don't think this is what he is referring to. A distribution is an instance (fixed parameters) of a family of distributions, but he is really talking about the difference between a RV and a given distribution (not a family of distributions). – samlaf Jul 17 '17 at 17:02
• Well, this is how I understood Blitzstein -- as far as I remember. Furthermore, the rest was covered by whuber in the comments. – Semoi Jul 17 '17 at 18:48