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I have a question about a random variable $S_n=X_1+X_2+...+X_n$ in the probability theory.

Assume that $X_k$ is a random variable on $\Omega$ for each $k$ and that each $X_k$ has the same distributions.

In probability theory, we study a random variable $S_n=X_1+X_2+...+X_n$. Since this sum is just a addition of a functions, $S_n$ must have the domain $\Omega$.

My question is the following: suppose we toss a coin $n$ times and let $X_k$ : $\Omega = \{H, T\} \to \mathbb{R}$ be random variables with $X_k(H)=1, X_k(T)=0$. Then $S_n=X_1+X_2+...+X_n=1+1+...+1=n$ and $S_n(T)=X_1(T)+X_2(T)+...+X_n(T)=0+0+...+0=0$. It does not mean anything that is useful! I know that $\frac{S_n}{n}$ must mean the average number of 'heads' in $n$ tosses of a coin. So I think the domain of $S_n$ should be the collection of n-tuples $\omega=(\omega_1, \omega_2, ..., \omega_n) \in \Omega$, where $ω_k$ is either $H$ or $T$ and $S_n(\omega)=X_1(\omega_1)+X_2(\omega_2)+...+X_n(\omega_n)$.

Can someone give me the right explanation about a domain of $S_n$?

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  • $\begingroup$ It depends on what you want to calculate/study with $S_n$. $\endgroup$
    – user99680
    Commented Nov 28, 2013 at 7:25
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    $\begingroup$ The ordinary choice is the one you made, the set of $n$-tuples. $\endgroup$ Commented Nov 28, 2013 at 7:29
  • $\begingroup$ Any space large enough to model the $n$ results would do, the product one you suggest at the end fits but any larger space would as well. You should soon see, if you learn more stochastic stuff, that specifying the underlying probability space is a BAD IDEA (what happens when you add a new head/tails?), and quite useless. $\endgroup$
    – Did
    Commented Nov 28, 2013 at 8:29

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In principle, one chooses a probability space: a set $\Omega$ (whose members are the individual "outcomes"), a $\sigma$-algebra $\Sigma$ of subsets of $\Omega$ (the events), and a probability measure $P$ on $\Sigma$. All the random variables you're interested in then correspond to $\Sigma$-measurable functions on $\Omega$.

In this case $H$ and $T$ label the possible outcomes for each individual coin-toss, but they don't capture the outcomes of the whole sequence of coin-tosses: heads on the first toss is not the same thing as heads on the second toss. Don't be misled by the fact that the same labels are used! To specify an outcome of the sequence of coin-tosses, you need to say which of $\{H, T\}$ occurred on each toss. Thus you take the set of $n$-tuples $\{H,T\}^n$.

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