Finding the probability space of the given experiment.

Question

Specify the probability space completely for the following experiment: tossing a fair coin till we see the first heads. Here is what I have done so far:

The sample space is simply $T^n H$ where $n$ can go from $0$ to infinity i.e. $\{T,TH,TTH,TTTH,\ldots\}$

Essentially the sample space is infinite. Let event $E_n$ be that head shows up after $n$ tails then $P(E_n)$ will be $(1/2)^{n+1}$ since we require $n$ tails and then a heads must show up.

How do I denote the sigma algebra since the sample space is infinite.

Also, I am under the impression that the sigma algebra will always be the power set of sample space. Is it correct? If so, then why use a special name sigma algebra and not simply call it the power set?

Thanks!

Answer 1

You have the right sample space, and you are right that if the $\sigma$-algebra were always the power set of the sample space, then there wouldn't be a point in having such a concept.

This is a discrete sample space. "Discrete" means the sum of the probabilities assigned to subsets of the sample space having just one member is $1$. In other words, all of the probability is accounted for by point masses. This particular discrete sample space also has no points with probability $0$. In discrete sample spaces with no points having probability $0$, the $\sigma$-algebra will always be the power set of the sample space.

Contrast this with the uniform distribution on the interval $[0,1]$. If $0\le a \le b\le 1$, then the probability assigned to the interval $[a,b]$ is the length $b-a$ of the interval. That means every subset with just one member has probability $0$. In this case the $\sigma$-algebra is the set of all Borel sets that are subsets of $[0,1]$. Borel sets are sets that can be reached by starting with intervals and closing under the operations of complementation and countable union.

Answer 2

The way I'd do it is as the $\sigma$-algebra generated by the sets of the form $\Delta(a_{1}, \ldots, a_{n}) = \{ (s_{k})_{k \in \mathbb{N}} \in \{T, H\}^{\mathbb{N}} : s_{k} = a_{k}, k = 1, \ldots, n\}$, that is, the set of sequences of tails and heads for which the first $n$ flips will agree with some prescribed sequence of $n$ outcomes, i.e. $\Delta(THHT)$ is the set of sequences beginning $THHT$, though what happens after is irrelevant. These are called cylinder sets, or cylinders, of rank $n$, and they generate a $\sigma$-algebra.

Your event is the disjoint union of the cylinders $\Delta(T^{k}H)$, each of measure $2^{-(k + 1)}$.