1. In a nuclear reaction, particles are emitted randomly so that in average 1000 particles are emitted per second. (a) Give three properties of the emission process that would allow to model it by a Poisson process. In what follows we shall model the number of particles emitted by a Poisson process. (b) (i) Compute the probability that two particles are emitted within 1 millisecond and none are in the next millisecond. (ii) Assume that at each emission, identically and independently from previous emissions, a particle a is emitted with probability p whereas a particle B is emitted with probability 1 - p. A random amount of energy is then released. The emission a particle releases in expectation 190 MeV for one a particle and 200 MeV for one B-particle. After 1 minute 116.10 MeV have been released. Give an estimate for the value of p, justifying your answer with the relevant theorems. (iii) Assume now that each emission the expectation and standard deviation of the released energy is respectively 200 MeV and 5 MeV. Compute the standard deviation of the energy released within 1 minute and 30 seconds. (c) For any 1 € (0,1] denote by L, the number of particles emitted from the beginning of the reaction up to time log (4), where t is counted in millisecond. Show that (L,1 € (0,1]) has same law as #{n 21:01...0, 2 t}, 0 < < 1 where U, U2,..., are i.i.d. uniform random variables on (0,1).