Did I mess up the Poisson-Gamma relationship?

Question

Let $X_1, X_2$ be independent, exponentially distributed random variables with mean 2. So $X_1+X_2=Z$ is gamma distributed with $\alpha=2$ and $\beta=1/2$.

I am trying to solve the probability that $X_1>3$ given that $X_1 +X_2>3$. This should be fairly straightforward, yet my answer differs from the book's answer.

I set up the solution as $$P[X_1>3]/P[Z>3]=e^{-1.5}/(1-P[Z<3])=e^{-1.5}/(1-P[Q<2]) $$ where $Q$ has a poisson distribution with mean equal to $x*\beta=3/2.$ As a result, I get $$ e^{-1.5}/(1-[1.5e^{-1.5}+e^{-1.5}])=.223/.442 = .50. $$

My book's answer is $.40.$ Does anyone see where I might have messed up?

Answer 1

\begin{equation} \begin{aligned} P(X_1 > 3 \mid X_1 + X_2 > 3) & = \frac{P(X_1 + X_2 > 3 \mid X_1 > 3)P(X_1 > 3)}{P(X_1 + X_2 > 3)} \\ & = \frac{P(X_1 > 3)}{P(X_1 + X_2 > 3)} \text{ since $ X_2 > 0 $ with prob. 1} \\ & = \frac{e^{-1.5}}{P(X_1 + X_2 > 3)} \text{ using Exp($\frac{1}{2}$) cdf} \\ \end{aligned} \end{equation}

You could easily evaluate $ P(X_1 + X_2 > 3) $ by conditioning on $ X_2 $ and applying the law of total probability. But if you insist on thinking about it in terms of a Poisson process, you can do the following.

In a Poisson process with rate $ \lambda $, the event where the sum of the first two inter-arrival times $ X_1 + X_2 $ is greater than 3 is precisely the event where 1 or fewer arrivals occurred in the time period up to 3.

Since $ \lambda = \frac{1}{2} $, the number of arrivals from time 0 to 3, which we'll call $ N(3) $, is distributed as Poisson($\frac{3}{2}$). Then, we have

$$ P(N(3) \leq 1) = P(N(3) = 0) + P(N(3) = 1) = e^{-3/2} + \frac{3}{2}e^{-3/2} = \frac{5}{2}e^{-3/2} $$

and therefore

$$ \frac{e^{-1.5}}{P(X_1 + X_2 > 3)} = \frac{e^{-3/2}}{\frac{5}{2}e^{-3/2}} = \frac{2}{5} = 0.4 $$

Answer 2

Z denotes the point in time when the second Poisson event occurred. Z>3 means that the second Poisson event occurred after time 3, and therefore it is equivalent to Q<2 (Q being the number of Poisson events up to time 3) and not to Q>=2 as you calculated. If you divide 0.223 by 1-0.442=0.558, you will get the correct answer 0.4.