As an algebraist who occasionally teaches introductory statistics and probability, this question has been on my mind for a while:
Can one use the Law of Large Numbers (LLN) to define the expected value of a random variable? If so: Why are we not doing this?
A little more precisely, I am rather convinced that the following proposition is a true restatement of the strong and weak LLN. (For which I assume the standard definition of real-valued random variables, of their distribution, and of independence. And for a sequence of random variables $(X_i)_{i \in \mathbb N}$, I abbreviate as usual $\overline X_n := \frac{1}{n}\sum_{i=1}^n X_i$.)
Proposition: Let $X$ be a random variable. Assume $X_1, X_2, ...$ are random variables which are independent, and all distributed identically to our $X$. Then:
- (strong law) There is at most one real number $c_s$ such that $P$-almost surely, $\lim_{n\to \infty} \overline{X_n} = c_s$.
- (weak law) There is at most one real number $c_w$ such that, for any $\varepsilon >0$, $\lim_{n\to \infty} P(|\overline{X_n} - c_w|<\varepsilon) =1$.
- If $c_s$ exists, then so does $c_w$, and $c_w=c_s$.
And then we define $E(X) := c_s$ (or $c_w$, which covers some non-classically integrable $X$) if it exists.
However, to prove existence quickly, I assume all of us would say, look, this is proven for $c :=$ the well-known $EX$ defined via (...), and it is easy to see that once it holds for that one number, it cannot hold for any other. By my algebraic upbringing, I feel like one should define an invariant not by (...), but rather by a kind of "universal property" like this. And I think that one version or another of the Law of Large Numbers is the natural candidate for such a "defining property". (For "meta-mathematical" background, cf. 1, 2.)
Because as abstract as this seems to non-algebraists, even as a teacher I think such "defining properties" are the best way to introduce concepts. Bright students will ask, okay, I accept your definition / formula, but why are we interested in this? What's the point of this number?
And just like when I introduce determinants in a linear algebra class, I would never ever start with the stupid formulae but by saying: "we want to attach one single number to a matrix which changes like this (namely, flip sign) if we flip rows, changes like that (namely, not at all) if we add a row to another, is 0 if the matrix is singular, is $1$ for the identity matrix, ..., and there is only one formula whose output satisfies all these useful properties ---
--- I want to say something like that for the expected value (and later, variance, standard deviation etc.)
Of course I first thought of more algebraic properties like linearity, monotonicity, the obvious norming ("if $X=c$ P-a.s., then $EX:=c$"). But then came the crucial $$E(\mathbf 1_{A}):=P(A) \qquad (*)$$ and I find this one is begging the question. Once one assumes $(*)$ and linearity, for random variables with finitely many values, everything seems determined, and after that, notwithstanding technicalities, it seems to be just a matter of monotonous / continuous expanding of the definition.
So the question becomes: Is there a line of argument which infers $(*)$ from the above proposition, without already using (*) in the proof of the proposition?
Maybe a variant of this question was asked here, but phrased very differently.
