Sample Space of a Geometric Distribution

Question

Let's say I have an experiment where I repeatedly toss a coin; each toss is independent. I would like to define a random variable $X: \omega \rightarrow R$. $X$ is the number of failures before the first success. I would like to visualize the sample space of this experiment, but I'm having trouble. Is the sample space an infinite set containing potentially infinitely long sequences?

Answer 1

There are always infinitely many valid ways to choose the sample space. Here are four natural ones:

1. The set of all infinite sequences of $T$ and $H$. The function $X$ gives the number of $T$s at the beginning of the sequence. In particular, $X(T,T,T,\dots)=\infty$.

2. The set of all infinite sequences, except for the all tails sequence. Now, $X$ is a finite number for all inputs.

3. The set of all finite sequences whose last entry is $H$ and whose other entries are all $T$. Here, we are taking examples $1$ or $2$ and ignoring some information.

4. The set of nonnegative integers. $X$ is the identity function. The probability measure is $P(\{n\})=(1-p)^{n}p$, where $p$ is the probability of heads. This is the same as example $3$, with the correspondence $T^nH\longleftrightarrow n$.

It does not matter whether we include the all tails sequence because the probability of it is zero. You can remove any probability zero event from a sample space without changing anything.