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Day #43 : Introduction to Functions of One Random Variable – 365 DoA

Function of one random variable (Borel)

Yep we’re getting a little complicated, but let’s see if we can make sense of this.

Now that we’ve looked at conditional probabilities we can talk about other things we can do with random variables. If you’ve been keeping up with us so far, then this shouldn’t be too crazy of an idea, really all we are going to do today is take a random variable and transform it somehow. Interested? Let’s go!*

First we should review what we mean by a random variable. If you’ve read our intro to the pdf (we reference that a lot, but it’s a good set of posts), you know by now that we say something along the lines of P(x) = some function. However, as I pointed out before, what we really mean is P(x) as X approaches x. When we talk about random variables and functions involving random variables, we are talking about this value X and the functions that discribes what happens to that random variable is our pdf and CDF.

However, what if we aren’t really interested in that variable, but a function of that variable? Well in that case what we are after is some transformation from one domain to another. If you are familiar with the Fourier transform, then this concept shouldn’t seem too crazy. Sometimes it’s easier to work with a function after we perform a transform and while we shouldn’t think of this as a transform from the time to frequency domain like we do when we talk about the Fourier transform (because it is not doing that), we should think of it in the same vane.

In an effort to keep this short, we will introduce the concept today and go over some examples next post (hence intro in the title, clever right?). So with any good transform there are a certain set of properties that it needs to have before we can say it’s useful. Those properties are:

  1. Its domain MUST include the range of our random variable (X).
  2. For every X mapped to Y the set Ry such that g(x) ≤ y must consist of the union of intersection of a countable number of intervals. Only then {Y ≤ y} is an event.
  3. The events {g(X) = ± ∞} must have zero probability

Basically we are saying for 1. that the new transformation needs to include the domain of X, which makes sense, we can’t use something that doesn’t include this range of values because the result would be the null set (like we saw yesterday).

The second point (2) is a little more complex, it is saying that the transform needs to be a Borel function. We haven’t (and may not) defined it so you may be a little confused by the definition. Assuming I understand this correctly (a big assumption), we are saying this transform is reversible and that for every point in our X space there exists a corresponding point in our Y space. For that last bit, all we are saying is that if there is a value in X then it exists in Y and it must be an event (have a value).

Lastly point three (3) is just saying that our functions are finite (again making the assumption that I understand this properly). This one if I understand correctly just sets our limits to the function so we don’t have Y(g(X) go off to infinity (which wouldn’t be useful).

All this to say this very simple thing, we have a value X that has some pdf f(X) attached to it, but we want to transform that to Y(f(X)) because we are interested in some function of f(X) not f(X) itself. To do this transform Y(f(X)) needs to be reversible, have a Y space value that corresponds to a X space value, and if the value is the null set in the X space, then it should be the null set in the Y space as well.

Summed up in one paragraph without all the fancy math, fun right? Of course it’s one thing to explain it that way, it’s another thing to show and prove it mathematically, hence all the extra language leading up to it. Next post we will go over an example, so if you’re confused (and I don’t see why you wouldn’t be after this) stick around and we can make sense of it together.

Until next time, don’t stop learning!

*My dear readers, please remember that I make no claim to the accuracy of this information; some of it might be wrong. I’m learning, which is why I’m writing these posts and if you’re reading this then I am assuming you are trying to learn too. My plea to you is this, if you see something that is not correct, or if you want to expand on something, do it. Let’s learn together!!

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