How is the sample space of a random variable defined?

Question

I was watching this example where the professor said that if we have 2 i.i.d RVs, both of them being binomial random variables, then we can only add them up if their sample spaces are the same. I am wondering what the meaning is of a sample space for a random variable?

Answer 1

The formal definition of a (real) random variable is a function $X:\Omega\to\mathbb R$, where $\Omega$ is the sample space. The sample space is, in informal terms, where the "randomness" of probability comes from.

For example, say you flip a fair coin $5$ times. Then the sample space $\Omega$ is the set of your possible outcomes, e.g. $HHHHH$, $THTTH$ etc.

A random variable $X$ in this space might e.g. be the number of heads you toss. So for example, $X(HHHHH)=5$.

It makes no sense to add two random variables with different sample spaces in the same way it makes no sense to add two functions with different domains. This is because just as with functions, we define $X+Y$ by $$(X+Y)(\omega)=X(\omega)+Y(\omega),$$ for $\omega\in\Omega$. Evidently we require $X$ and $Y$ to have the same sample space.

As an example, consider $X$ and $\Omega$ as defined by the coin tosses, and consider a new random variable $X':\Omega'\to\mathbb R$. The sample space $\Omega'$ is the set of points on Earth, and $X'$ is the current temperature at a point on the Earth. What is $X+X'$?

The situtation can be salvaged though. Suppose $X$, $Y$ are real random variables with sample spaces $\Omega_X$, $\Omega_Y$. Define a random variable $Z:\Omega_X\times\Omega_Y\to\mathbb R$ by $$Z(\omega_X,\omega_Y)=X(\omega_X)+Y(\omega_Y).$$ Then the projections $\bar{X}:(\omega_X,\omega_Y)\mapsto X(\omega_X)$ and $\bar{Y}:(\omega_X,\omega_Y)\mapsto Y(\omega_Y)$ have the same distributions as $X$ and $Y$ respectively, and clearly $\bar{X}+\bar{Y}=Z$ is defined. In fact, $\bar{X}$ and $\bar{Y}$ are even independent.