# Questions regarding this derivation of the Poisson Distribution from exponential densities

On page 217 - 218 of the pdf of this book, the author derives the Poisson Distribution using gamma and exponential densities. The author defines $$S_n$$ to be the sum of a sequence of independent exponentially distributed random variables ($$X_1 + X_2 .... + X_n$$) with parameter $$\lambda$$ and $$Y$$ to count the number of emissions in a time interval $$t$$, and then states the following:

$$P(Y = n) = P(S_n < t \space and \space S_{n+1} > t) = P(S_n < t) - P(S_{n+1}

The reasoning being that $$S_{n+1} < t$$ is a subset of the event $$S_{n} < t$$

1. I don't understand why $$P(Y = n)$$ is equal to $$P(S_{n} < t < S_{n+1})$$, why the event $$S_{n+1} < t$$ is a subset of $$S_{n} < t$$ (and not the other way around?), and why we take the difference of the two?

2. Is the $$\lambda$$ defined for the sequence of exponential densities, in the gamma density (used to model $$S_n$$), and the final Poisson distribution, all the same?

3. To simulate a Poisson random variable ($$W$$) with parameter $$\lambda$$ using exponential densities, the author proposes that $$-\frac{1}{\lambda}log(rnd)$$ (which simulates an exponential density with parameter $$\lambda$$) be run sequentially until $$S_{n} < 1 < S_{n+1}$$ and the $$n$$ returned is the simulated value for $$W$$. Would I replace the $$1$$ with $$t$$ if I want to simulate a Poisson random variable with parameter $$\lambda t$$?

If $$Y(t) = \sum_{n=1}^\infty \mathsf 1_{(0,t]}(S_n)$$ is the counting process associated with the renewal times $$\{S_n\}$$, then by definition $$\{Y(t) = n\} = \{S_n\leqslant Y(t) < S_{n+1}\}.$$ This is simply because $$n$$ renewals have occured by time $$t$$, and the $$(n+1)^{\mathrm{th}}$$ renewal has yet to occur. As for $$\{S_{n+1}, we have $$\{S_{n+1} since $$\mathbb P(X_{n+1}\geqslant 0)=1$$.
$$\lambda$$ is a constant, and is the same everywhere.
To simulate a Poisson random variable with rate $$\lambda t$$, you would use $$-\frac1{\lambda t}\log U$$ where $$U$$ is uniformly distributed over $$(0,1)$$.
• 1. Is it fair to argue that if $S_{n+1} \leq t$ means $S_n \leq t$ but the reverse is not always true, hence the event $S_n \leq t$ includes the event where $S_{n+1} \leq t$ and thus the latter is a subset of the former? 2. To generate a Poisson r.v. $W$, what if I tweak the procedure in my book: generate a sequence of exponential r.v. each with $-\frac{1}{\lambda}log(rnd)$, keep track of subtotals $S_k$ (sum of the exponential r.v.), and if $W$ has parameter $\lambda t$ instead of $\lambda$, I take $n$ such that $S_n < t < S_{n+1}$ instead of $S_n < 1 < S_{n+1}$ per my book. Is this valid? – Yandle Jan 21 '20 at 18:50