Poisson processes introduction - why $\{N(t) \geq n\} = \{S_n \leq t\}$?

Question

I recently got introduced to Poisson Processes and while we were preparing necessary mathematical tools to do it, something along the following lines was in the notes of our professor.

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Define 0 < $S_1$ < $S_2$ < $\dots$ < $S_n$ < $\dots$ as a time of arrival of event $S_1$, $S_2$, $\dots, S_n, \dots$ and let $N(t) = \sup\{n > 0: S_n \leq t\}$ (how many events occured until time $t$). Notice that $\{N(t) \geq n\} = \{S_n \leq t\}$.

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We then used $\{N(t), t \geq 0\}$ as our first Poisson process (with $S_n$ being $\sum_k T_k$ where $T_k$ i.i.d. exponential).

I am trying to wrap my head around $\{N(t) \geq n\} = \{S_n \leq t\}$. I remember my lecturer saying something along lines "we can infer $t$ from $n$ and $n$ from $t$".

Why are these sets equal? What are their items' "types"? If I understand correctly $\{S_n \leq t\}$ contains all events from 1 to $n$ that occured until time $t$. The second set however, $\{N(t) \geq n\}$ contains just natural numbers.

I appreciate your help.

Answer 1

These are events, not sets of numbers.

$N(t)$ is an integer-valued random variable that counts the number of arrivals (these are called "events" in your notes, but I won't use this word to avoid confusion with the above usage of the word "event.") in the time interval $[0,t]$. The event $\{N(t) \ge n\}$ contains all outcomes where "there are at least $n$ arrivals in the time interval $[0,t]$."

$S_n$ is a real-valued random variable that is the time of the $n$th arrival. The event $\{S_n \le t\}$ contains all outcomes where "the $n$th arrival occurs before time $t$."

Do you now see why these two events are equivalent?

Answer 2

$\{N(t) \ge n\}$ is not a set of natural numbers, it is an event (a measurable subset of the sample space). Same with $\{S(n) \le t\}$.

$N(t) \ge n$ means there are at least $n$ arrivals in the time interval $[0,t]$. That's equivalent to saying that the $n$'th arrival happens by time $t$, i.e. $S_n \le t$.

Answer 3

The probability that $n$ or more events occurred by time $t$ (aka $\{N(t) \geq n\})$is the same as the probability that the nth event arrived before time $t$ (aka $\{S_n \leq t\})$.

Suppose for a moment that the nth event ($S_n$) arrived after time $t$. Then we would have less than n events at time $t$ which is not enough, so we need to have $Sn<t$.