# Let $N(t)$ be a Poisson process, compute $P\{N(s)=1,N(t)=2\}$ for any $0\leq s<t$

I got the answer as $$\lambda^2e^{-\lambda t}s(t-s)$$ using the properties like independent increment and stationary increments. But I can't seem to understand the steps in the solution of the book.

For reference, the definition followed in the book is, let $$\eta_1,\eta_2,\ldots$$ be a sequence of independent random variables,each having the same exponential distribution of rate $$\lambda$$.We put

$$\xi_n=\eta_1+\ldots+\eta_n$$

$$N(t)$$, where $$t\geq0$$, is a Poisson process if $$N(t)=max\{n:t\geq\xi_n\}$$

Solution: Using the fact that $$\eta_1,\eta_2,\ldots$$ are independent and exponentially distributed, we obtain

$$\begin{array} {lcl} P\{N(s)=1,N(t)=2\} & = & P\{\xi_1\leq s<\xi_2\leq t<\xi_3\} \\ & = & P\{\eta_1\leq s<\eta_1+\eta_2\leq t<\eta_1+\eta_2+\eta_3\}\\ &=& \displaystyle\int_{0}^{s}P\{s

I understood the first two lines.For the next step, I thought maybe they used the PDF of the exponential distribution of $$\eta_1$$ and took the PDF as function of $$u$$. So the replacing of $$\eta_1$$ by $$u$$. But can we just do that? Didn't it become a probability of an event by the second line? So how can we consider a single inequality in it?

I'm trying to self learn Stochastic Process.So my knowledge isn't the best in this. Any help would be appreciated.

Btw I solved the problem just like https://math.stackexchange.com/questions/4356860/let-nt-be-a-poisson-process-calculate-pns-neq-nt-and-pns-0-nt

• They are using the law of total probability and using the independence of the $\eta_j$'s. Commented Jun 19, 2023 at 7:44

It involves the knowledge of multiple integrals.

Define $$D=\{(\eta_1,\eta_2,\eta_3): \eta_1\leq s<\eta_1+\eta_2,

and $$A(u)=\{(\eta_2,\eta_3): s, where $$u \leq s$$.

Given that $$\eta_1,\eta_2,...$$ are mutually independent, the PDF $$f(\eta_1,\eta_2,\eta_3)=P(\eta_1)P(\eta_2)P(\eta_3)$$.

We have

\begin{aligned}P(\eta_1

Above describes the deduction process from the second line to the third line. The same reasoning applies to deducing the result from the third line.

• It kind of makes sense.Guess I need to revise some concepts for this
– A Y
Commented Jun 19, 2023 at 17:05