Identity for the CDF of Poisson Random Variable 
My question is how does the left hand side equation for part a equal to the probability $P(W_n >1)$ where $W_n$ is a Poisson process on the unit interval with mean $\lambda$? The method I used to prove part a is using the gamma and beta functions.

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*The waiting time until the $n$th event can be written as $W_n = T_1 + T_2 + \cdots + T_n$, where $T_i$ is the waiting time between the $(i-1)$th event and the $i$th event (and where $T_1$ is the waiting time until the first event).

*For a Poisson process with rate $\lambda$, the $T_i$ are i.i.d. exponential random variables with rate $\lambda$.

*The sum of $n$ independent $\text{Exponential}(\lambda)$ random variables can be shown to follow a $\text{Gamma}(n, \lambda)$ distribution, whose density appears in the left-hand side of the equation you are asking about.

