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For a nonnegative integer-valued random variable, its expected value is $\mathbb{E}N={\sum }_{r=0}^{\infty }r{p}_r$, allowing the value $+\infty$. Since $r={\sum }_{n=1}^{r}1$, rearranging nonnegative summands gives the tail-sum formula for expectation:No integrability assumption is needed for the rearrangement. The same formula holds for an extended nonnegative integer-valued variable by the monotone convergence theorem, using $N={\sum }_{n\ge 1}{1}_{\{N\ge n\}}$.

For the first ascent in independent continuous observations, continuity makes ties probability zero. By order symmetry of independent random variables, the first $m$ observations have each of their $m!$ strict orderings with equal probability. For $n\ge 2$, $N\ge n$ means exactly that the first $n-1$ observations are decreasing, soSubtracting consecutive tails giveswith probability zero for $r=0,1$. The events $\{N\ge n\}$ decrease to $\{N=\infty \}$; continuity of probability under decreasing intersections gives $P(N=\infty )={\lim }_{n}1/(n-1)!=0$. Finally,The first observation cannot itself end the run; this is why the tail includes the separate term $P(N\ge 1)=1$.

The independent and identically distributed random variables have a continuous probability distribution, so any tie has probability zero. Their joint probability density is unchanged by a permutation of their indices: this is order symmetry of independent random variables. The $n!$ strict orderings are disjoint and exhaust the sample space apart from the zero-probability ties. Each ordering therefore has the same probability, and their probabilities sum to one.