Example of two random variables

Question

I can't understand a concept of a random variables. If I have a random variable X which maps tosses of fair coin to (0,1) where 0 for heads and 1 for tails. How can there exist another one random variable? My random variable X already matches all possible tosses to their outcomes.

Answer 1

A random variable is a measurable function $X:\Omega\rightarrow E$. In your example you can take $\Omega:=\{\text{head},\text{tail}\}$ and $E=\{0,1\}$. Then one can define $$X(\omega)=\begin{cases} 1, & \text{if } \omega = \text{heads}, \\[6pt] 0, & \text{if } \omega = \text{tails}. \end{cases}$$ But on the other hand you could also define $$Y(\omega)=\begin{cases} 1, & \text{if } \omega = \text{tails}, \\[6pt] 0, & \text{if } \omega = \text{heads}. \end{cases}$$ Then clearly $X$ and $Y$ are different indeed $$\begin{align}P(X=Y)&=P(\{\omega\in \Omega: X(\omega)=Y(\omega)\})\\&=P(\emptyset)\\&=0\end{align}$$ so $X\neq Y$ a.s.

Therefore we have now found a second random variable. This shows you that a random variable only model a situation but this does not happen uniqly.

Answer 2

The sample space is the set of all fully-specified outcomes/timelines/worlds that can result from the experiment(s) being considered. A random variable is a function that tells you a piece of information about the outcome.

For example, if your experiment consists of tossing two coins, you could have two random variables $X$ and $Y$, each telling you the result of one of the coin tosses, and a third random variable $Z$ telling you the number of heads that came up. (In this example, the sample space needs to include an outcome for every possible result of the pair of coin tosses, so you could define it as $\Omega=\{hh,ht,th,tt\}$ and define $X,Y,Z$ by specifying their values on each of the four outcomes, e.g. $X(hh)=Y(hh)=1$, $Z(hh)=2$).

You can also (somewhat pointlessly) define different random variables that represent the same information (e.g. $Z$ and $Z+55$) or random variables that give no information at all (e.g. the function whose value is "banana" for every outcome).