We define an infinite sequence of independent random variables $U1, U2, U3, ...$, uniformly distributed across the open interval $(0,1)$.
We also define another random variable, $N=\min \{ n\in \mathbb{N} | \prod ^{n+1}_{j=1}U_{j} < e^{- \lambda}\}$
How is N distributed?
What I've done so far:
To my understanding, we need to find $P(N=k)$:
(1). As for the first step, I was trying to understand what is the set that N is distributed from. So all set members follow $$\begin{aligned}\prod ^{n+1}_{j=1}U_{j} <e^{-\lambda }\\ \downarrow \\ \ln \left( \prod ^{n+1}_{j=1}U_{j}\right) <-\lambda \\ \downarrow \\ \sum ^{n+1}_{j=1}\ln \left( U_{j}\right) <-\lambda \\ \downarrow \\ -\dfrac{1}{\lambda }\sum ^{n+1}_{j=1}\ln \left( U_{j}\right) > 1 \end{aligned}$$
Denotating $-\dfrac{1}{\lambda }\ln \left( U_{i}\right) = X_{i}$, I found that for any $i\in \left[ n\right]$, $X_{i}$ is distributed exponentially with expected value $\lambda$. So the sum $X1 + X2 + X3 + ... $ follows a Gamma distribution with parameters $(n, \lambda)$.
(2). If $N=k$, then for any $i\in \left[ k - 1 \right]$, $\prod ^{i+1}_{j=1}U_{j}\geq e^{- \lambda}$, and for any $i \geq k$, $\prod ^{i+1}_{j=1}U_{j} < e^{- \lambda}$.
(3). I wanted to calculate $P(N=k)$ using the law of total probability, but I found it difficult to understand what's $P(min \; is \; i)$ due to (2)
Would appriciate some help! Thanks!