# What is the formalism that allows Random Variables to be treated algebraically like real or complex numbers?

We all know that if we have a variable x, then there is a meaning to - for example - $$y=e^x$$.

And we all know how to manipulate that algebraically and to do calculus. For example, if

$$y_1=e^{x_1}$$

and

$$y_2=e^{x_2}$$

then

$$y_1y_2 = e^{x_1+x_2}$$

But random variables are a very different beast. Ultimately we all know intuitively what we mean when we draw a PDF or, more accurately, the CDF which is itself an algebraic function mapping $$[-\infty, \infty]$$ to $$[0,1]$$ as part of that defines a random variable.

But what is the actual math that makes that definition of an RV actually act algebraically, so that, as above, we can happily write things like

$$Y_1=e^{X_1}$$

and

$$Y_2=e^{X_2}$$

then

$$Y_1Y_2 = e^{X_1+X_2}$$

where $$X_1$$ and $$X_2$$ are random variables - and get away with it???

• If you are worried about measurability issues, there are basic theorems that say the product of two measurable functions is measurable, and so on. Otherwise, the fact $Y_1(\omega)Y_2(\omega)=e^{X_1(\omega)+X_2(\omega)}$ for all $\omega$ follows by substituting the definitions of $Y_i(\omega)$ (as the answers below indicate) and the PDF/CDF has nothing to do with it...this is a more basic fact about substitution. – Michael May 22 at 2:22

Formally $$X_i$$ are real valued (measurable) functions i.e $$X_i\colon \Omega\to \mathbb{R}$$ for $$i=1,2$$ where $$\Omega$$ is the sample space. So $$Y_i=e^{X_i}$$ for $$i=1,2$$ as an equality of functions means that $$Y_i(\omega)=e^{X_i(\omega)}$$ for all $$\omega\in \Omega$$. Hence since $$X_i(\omega)$$ is a real number it follows using the first part of your question that $$Y_1(\omega)Y_2(\omega)=e^{X_1(\omega)}e^{X_2(\omega)}=e^{X_1(\omega)+X_2(\omega)}$$ for all $$\omega$$ whence as functions $$Y_1Y_2=e^{X_1+X_2}.$$
• It’s probably worth pointing out that one can also consider equality in distribution, i.e. $Y$ is a random variable with distribution equal to the distribution of $e^X$, as well as almost sure equality. – icurays1 May 22 at 0:26
• @eSurfsnake: Random variables need not take values in the interval $[0,1]$, the values can be anything. You might be confusing it with probabilites, which must of course be in the interval $[0,1]$. An event is a (measurable) subset of $\Omega$, and the probability of an event is its measure, which is at most $1$, since a probability space $\Omega$ by definition has measure $1$. – Hans Lundmark May 22 at 5:57
• yes, to me an RV is tightly bound with a probability - otherwise, what is the point of probability theory? But, am I right to assume that the abstract concept of a measurable function means precisely that if I have and RV, X, with its CDF, then a "measurable function" g(X) maintains certain logical relations; i.e., that a domain of measure $\mu$ on X maps to a range of meaningful measure such that no contradictions occur between X and g(x)? – eSurfsnake May 23 at 6:04
Formally a random variable is simply a measurable function $$X:\Omega\to \mathbb R$$, where $$\Omega$$ is a sample space equipped with some sigma algebra, $$\Sigma$$, containing the events we are interested in, and a probability measure $$\mathbb P$$. We call the triple $$(\Omega, \Sigma, \mathbb P)$$ a probability space. Thus if $$g:\mathbb R\to \mathbb R$$ is some "regular" function, $$g\circ X:\Omega\to \mathbb R$$ is simply the function mapping $$\Omega\ni\omega\mapsto g(X(\omega))$$. Thus if $$X_1$$ and $$X_2$$ are both random variables on the same probability space it is easy to talk about things like $$g\circ(X_1+X_2)$$. This is simply the map $$\omega\mapsto g(X_1(\omega)+X_2(\omega))$$.
For an example we can consider the most simple of experiments: flipping a fair coin twice. Then our sample space is $$\Omega=\{HH,HT,TH,TT\}$$, and our $$\sigma$$-algebra is simply the power set. The most natural probability measure is the one assigning a value of $$1/4$$ to each singleton in $$\Omega$$. Suppose we win a bet if in the experiment two heads occur. We can then capture this by the random variable $$X:\Omega\to \mathbb R$$ given by $$X(HH)=1$$ and $$X$$ is $$0$$ on all other elements of $$\Omega$$. Thus we see $$X$$ is simply a function, just not from $$\mathbb R$$ to itself.
Often in statistics we actually don't see $$\Omega$$ written out explicitly, because it often consists of weird things, like the population of the earth, but it is always exists underneath when we're doing probability calculations. In fact for a r.v. $$X$$ the CDF, $$F_X$$ is defined in terms of the probability space as follows $$F_X(x)=\mathbb P\{\omega\in \Omega:X(\omega)\leq x\}.$$