Lifetime of a bulb is distributed $\operatorname{exp}(\lambda)$. When one light bulb is burned we replace it immidietely. Let $N_t$ be the number of bulbs we've used by time $t$. Prove that $N_t \sim \operatorname{Poi}(\lambda t)$.
Where is the mistake in my solution?
For all $i\in \mathbb{N}$, let $X_i \sim\operatorname{exp}(\lambda)$ be the lifetime of bulb #$i$. $(X_i)$ are independent. $$ \\ \mathbb{P}(N_t=k)=\mathbb{P}(t-1<X_1+...+X_k\leq t) \ \\= \dots = (t\lambda)^{k-1}\cdot e^{-\lambda t} (e^{-\lambda}-1) $$
Mistakes:
1) In first line on RHS you use $n$ instead of $k$.
2) $\{N_t=k\}\neq\{t-1<X_1+\cdots+X_k\leq t\}$
What we do have is $\{N_t= k\}=\{X_1+\cdots+X_k\leq t<X_1+\cdots+X_k+X_{k+1}\}$.
It is handsome to make use of: $$P(N_t=k)=P(N_t\geq k)-P(N_t\geq k+1)$$