I'm trying to prove a common result for the Poisson process but I'm stuck.
Given $T_i$ are i.i.d. $Exp(\lambda)$ random variables (where $\lambda$ is the rate) that represent the duration of arrival events and $N(t) = \textrm{max} \{ n: \sum_{i=1}^{n}T_i \leq t \}$ is the number of said events within the interval $[0, t]$, prove that the mean of $N(t)$ is $\lambda t$.
Intuitively it makes a lot of sense: if the average duration between the events is 2s, then in a 10 second duration then the average # of events would be 5. But I wanted to see it mathematically.
My strategy was to use the exponentially distributed durations to get a mathematical expression that I can say resembles a Poisson with mean of $\lambda t$. I ended going down the path using the fact that $Y_k = T_1 + T_2 + \ldots T_k \sim Gamma(k,\lambda)$ and getting a product of Gamma CDFs:
\begin{align} P(N(t) = k) &= P(Y_k \leq t, Y_{k+1} > t) \newline &= P(Y_k \leq t) (1 - P(Y_{k+1} < t)) && (T_i\text{ are i.i.d.}) \newline &= \bigg( e^{-\lambda t}\sum_{i=k}^{\infty}\frac{(\lambda t)^i}{i!} \bigg) \bigg( e^{-\lambda t}\sum_{i=0}^{k}\frac{(\lambda t)^i}{i!} \bigg) && \text{from Wikipedia} \end{align}
which looks kind of close but I couldn't get further.
