Why isn't there a bound on the waiting time for the first occurrence in Poisson distribution? From my book:

Let $W$ denote the waiting time until the first occurrence during the observation of a Poisson process in which the mean number of occurrences in the unit interval is $\lambda$.
...Thus, when $w>0$, the pdf of $W$ is
$$F'(w) = f(w) = \lambda e^{-\lambda w}$$

My question is, why does it say "when $w>0$ as opposed to "when $1>w>0$? Since we are on a unit interval, I would've thought $W$ is bounded above by $1$.
 A: Read the wording extra carefully! The text states that lambda is the "...mean number of occurrences in the unit interval...". It does not state that our distribution is bounded by the unit interval.
We could define lambda over any interval we wanted. This way though lambda is directly interpretable as the rate of occurrence, so it's a convenient one to use. 
A: We are not on the unit interval. The waiting time $W$ for the first occurrence can have any nonnegative real number value; it is the number of occurrences $N$ in $(0,1]$ that is a Poisson random variable with mean $\lambda$ and hence probability mass function 
$$p_N(k) = e^{-\lambda}\frac{\lambda^k}{k!}, ~ k = 0, 1, 2, 3, \ldots$$
Note that $\lambda$ is also known as the parameter of the Poisson random variable $N$.
More generally, the number of occurrences in $(0,T]$ is a Poisson random variable $N_{(0,T]}$ with parameter $\lambda T$.  Thus, the probability that the waiting time $W$ for the first arrival is greater than $T$ is
$$P\{W > T\} = P\{N_{(0,T]} = 0\} = e^{-\lambda T}.$$
But, $P\{W > T\} = 1 - F_{W}(T)$ and so $F^\prime(T) = f_W(T) = \lambda e^{-\lambda T}$ for $T > 0$.
