I want to know how to go about proving that the Good-Turing estimator has a total probability of $1$. I have seen this proof (page 2) but I found unclear the first step:
$\sum_j \theta[j] = \sum_r \theta[r]N_r = \frac{1}{N}\sum \left[(r+1) \frac{N_{r+1}}{N_r}\right]N_r$
$\theta[j]$ is the probability of having an $n$-gram, $\theta[r]$ is the probability of a $n$-gram occurring $r$ times and $N_r$ is the number of $n$-grams that occur $r$ times.
Since $\sum_r(r+1)N_{r+1}$ = $\sum_r rN_r$, it's more or less straightforward that it actually sums $1$. However, as I said, I don't understand the first part: $\sum_j \theta[j] = \sum_r \theta[r]N_r$. What is going on there?
Thanks in advance.