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My professor wrote the following statement down as a starting point for a problem that I'm working on, and I don't quite see where it's coming from.

We are told that $\{N(t); t \ge 0\}$ is a simple renewal process, where the lengths of its interrenewal intervals have cdf $F$. The interrenewal lengths are denoted by $\{X_k; k\ge 0\}$. The statement that my professor made was $$\mathbb{E}[N(t)] = \int_{[0,\infty)} \mathbb{E}[N(t) \ |\ X_1 =s] dF(s)$$ I understand that for any random variable $X$ with cdf $F_X$, $$\mathbb{E}[X] = \int_{(-\infty,\infty)} x dF_X(x)$$ I just don't quite see how we get that conditional expectation inside the integral. I feel like I'm missing some fundamental fact about conditional expectation, rather than some fact about renewal theory. Can anyone help me see what I'm missing?

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The fundamental fact of conditional expectation which you are missing is the law of total expectation, also known as the tower property of conditional expectation.

Starting out with a simpler case, consider a matrix with entries $a_{ij}$, where $1\leq i,j\leq n$. You can take the average of all the entries "at once" and arrive at $$ \textrm{average}=\frac{1}{n^2}\sum_{i,j=1}^n a_{ij}. $$ Or, you can first average the columns and then average the averages: $$ \textrm{average of column $i$} = \frac{1}{n}\sum_{j=1}^n a_{ij}, $$ $$ \textrm{average of column averages} = \frac{1}{n}\left(\sum_{i=1}^n\frac{1}{n}\sum_{j=1}^n a_{ij}\right). $$ But of course, the two averages turn out to be the same - and this is the law of total expectation (in the discrete case). Indeed, you can think of the column averages as the condition expectation with respect to the column number.

The intuition behind this is that conditional expectation is a limited form of averaging where you keep some quantity random and average out the randomness in some other quantity. But then when you take the second expectation, you are averaging out all the remaining randomness.

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Actually, only the definition of conditional expectation is being used. This question should provide some help: Definition of Conditional expectation of Y given X.. Otherwise, my suggestion is to study up on the general definition of conditional expectation(tips on references are provided in the link).

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