The Wald's equation even at its simplest form as stated below simplifies many problems of calculating expectation.
Wald's Equation: Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of real-valued, independent and identically distributed random variables and let $N$ be a nonnegative integer-value random variable that is independent of
the sequence $(X_n)_{n\in\mathbb{N}}$. Suppose that $N$ and the $X_n$ have finite expectations. Then
$$
\operatorname{E}[X_1+\dots+X_N]=\operatorname{E}[N] \cdot\operatorname{E}[X_n]\quad \forall n\in\mathbb{N}\,.
$$
I am looking for an intuitive explanation of Wald's equation without using the optional stopping theorem.
I'm not interested in explanations for the error
$
\operatorname{E}[X_1+\dots+X_N]=N\cdot \operatorname{E}[X_1]
$
or explanations to discuss only the hypotheses.
We could have, for exemple, a function $\varphi$ of two or more variables such that $\operatorname{E}[X_1+\dots+X_N]=\varphi\big(\operatorname{E}[N]\, ,\,\operatorname{E}[X_n]\big), \quad\forall n\in\mathbb{N}\,$.
The question then becomes for what reason $\varphi(x,y)$ equals $x\cdot y$?
More generally we could have two linear functional $F : L^1(\Omega,\mathcal{A},P)\to \mathbb{R}$ and $G : L^1(\Omega,\mathcal{A},P)\to \mathbb{R}$ such that $\operatorname{E}[X_1+\dots+X_N]=\varphi\big(\operatorname{F}[N]\, ,\,\operatorname{G}[X_n]\big), \quad\forall n\in\mathbb{N}\,.$ So the question would be for what reason $\operatorname{F}=\operatorname{G}=\operatorname{E}$ and $\varphi(x,y)$ equals $x\cdot y$?
The interest is on the intuition of the equation. An answer based on a good example will be very welcome.
Thanks in advance.
Best Answer
One simple intuitive explanation is that $$ \mathbb{E}[X_1+\cdots+X_N | N=n] = n\mathbb{E}[X_1], $$ so it follows that $$ \mathbb{E}[X_1+\cdots+X_N] = \mathbb{E}[\mathbb{E}[X_1+\cdots+X_N|N]] = \mathbb{E}[N \mathbb{E}[X_1]] = \mathbb{E}[N] \mathbb{E}[X_1]. $$ This works because they are independent, so you just take $N$ copies of the same r.v. The identity is not quite as trivial when $N$ is a stopping time.