1. ## Poisson processes

A radioactive source emits particles according to a Poisson process with rate $\lambda=2$ particles per minute. What is the expected time of emission of the second particle given that three particles have been emitted in the first four minutes?

A radioactive source emits particles according to a Poisson process with rate $\lambda=2$ particles per minute. What is the expected time of emission of the second particle given that three particles have been emitted in the first four minutes?
Use Bayes theorem to find: $P(t_2|t_4\le 4)$ then the expected time of emmision of the second particle is:
$\displaystyle \int_0^4 t_2 P(t_2|t_4\le 4)\ dt_2$