Sum of probabilities equal expectations

Question

On page 18 of Norris book he writes the equality

$$ \sum_n^{\infty} P_i(H^A\geq n) =E_i(H^A) $$

And he previously noted the expectation to equal

$$ E_i(H^A)=\sum_n^{\infty}n P(H^A) $$

I am not sure how it can be formally shown that the first equality holds and how to prove that it is also equivalent for the second line.

Here

$H^A = inf\{n \geq 0: X_n \in A\}$

Is the hitting time of set A and X_n is the relevant Markov Chain.

Answer 1

If $X\ge 0$ is non-negative random variable supported on the integers we have that $$ EX=\sum_{n=0}^\infty P(X>n) $$ To see this, note that $$ EX=\sum_{n=1}^\infty n P(X=n)=\sum_{n=1}^\infty\left(\sum_{i=1}^n1\right ) P(X=n)=\sum_{i=1}^\infty\sum_{n=i}^\infty P(X=n)=\sum_{i=1}^\infty P(X>i-1) $$ where the interchanging of order of summation is allowed because we are dealing with non-negative series.