(4.4) For a nonnegative integer-valued random variable $N$, show that
$$ \E{N}=\sum_{i=1}^{\infty}\pr{N\geq i} $$
Hint: $\sum_{i=1}^{\infty}\pr{N\geq i}=\sum_{i=1}^{\infty}\sum_{k=i}^{\infty}\pr{N=k}$. Now interchange the order of summation.
Solution Note that $\pr{N\geq i}=\sum_{j=i}^{\infty}\pr{N=j}$. Hence
$$ \sum_{i=1}^{\infty}\pr{N\geq i}=\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\pr{N=j} $$
Note that the double sum is the sum over all different integer pairs $(i,j)$ such that
$$ 1\leq i\leq j<\infty $$
Hence
$$ \sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\pr{N=j}=\sum_{j=1}^{\infty}\sum_{i=1}^{j}\pr{N=j} \tag{4.4.1} $$
since the double sum on the right side is also the sum over all different integer pairs $(i,j)$ such that $1\leq i\leq j<\infty$. Also note that
$$ \sum_{i=1}^{j}\pr{N=j}=\pr{N=j}\sum_{i=1}^{j}1=j\wts\pr{N=j} $$
Hence
$$ \sum_{i=1}^{\infty}\pr{N\geq i}=\sum_{i=1}^{\infty}\sum_{j=i}^{\infty}\pr{N=j}=\sum_{j=1}^{\infty}\sum_{i=1}^{j}\pr{N=j}=\sum_{j=1}^{\infty}j\wts\pr{N=j}=\E{N} $$
It’s helpful to see a finite version of 4.4.1 is:
$$ \sum_{i=1}^2\sum_{j=i}^2a_{i,j}=a_{1,1}+a_{1,2}+a_{2,2}=\sum_{j=1}^2\sum_{i=1}^ja_{i,j} $$
It’s also helpful to note the similarity to Lemma 2.1, ch.5, p.191-192, as well as my notes. The proposition statements and their proofs parallel one another almost exactly. The only difference is discrete vs. continuous.
Also note that for 4.4.1 to be valid, we should consider Tonelli’s or Fubini’s Theorems.