The amount of earthquakes that happen at island X follows the Poisson process with mean 2 . Given that 2 earthquakes have happened in this year, find the probability both the earthquakes happen between the January-March period. Assume 1 month=30 days and 1 year=360 days.
Here is my attempt: $P(N(1/4)-N(0)=2|N(2)-N(0)=2)=\frac{P(N(1/4)-N(0)=2)}{P(N(2)-N(0)=2)}=e^{-1,5}\frac{1}{16}$ However, it's not in the options . Is my approach wrong?
There is an error in the first equality. Recall the definition of a conditional probability is $$ \operatorname{P}[A|B]:=\frac{\operatorname{P}[A\cap B]}{\operatorname{P}[B]}, $$ when $\operatorname{P}[B]>0$. Therefore, combing this with the independent increment property of the Poisson process $N$, $$ \operatorname{P}[N_{1/4}-N_0=2|N_2-N_0=2]=\frac{\operatorname{P}[N_{1/4}-N_0=2]\operatorname{P}[N_2-N_{1/4}=0]}{\operatorname{P}[N_2-N_0=2]}. $$
See the other answer how to solve it in the original way. If you know more about Poisson processes, there is a simpler solution. Conditional on the amount of points in an interval, the points are independently and uniform distributed, so the probability for both of them to fall into $[0,\frac14]$ is $\left(\frac18\right)^2$.
