91影视

Radioactive particles reach a target according to a Poisson process with unknown rate\(\beta \). In Exercise 22 of Sec. 5.7,we have to find the conditional distribution of\(\beta \)after observing the Poisson process for a certain amount of time.

Suppose that radioactive particles strike a target according to a Poisson process with rate\(\beta \), Let\({Z_k}\)be the time until the k^th particle strikes the target for k=12.... What is the distribution of\({Z_1}\)? What is the distribution of\({Y_k} = {Z_k} - {Z_k} - 1\)for\(k \ge 2\)?

Although the random variables defined at the end of Example 5.7.8 look similar

to those in Theorem 5.7.11, there are major differences. In Theorem 5.7.11, we were observing a fixed number n of lifetimes that all started simultaneously.

The n lifetimes are all labelled in advance, and each could be observed independently of the others. In Example 5.7.8, there is no fixed number of particles being contemplated, and we have no well-defined notion of when each particle 鈥渟tarts鈥� toward the target. In fact, we cannot even tell which particle is which until after they are observed. We merely start observing at an arbitrary time and record each time a particle hits. Depending on how long we observe the process, we could see an arbitrary number of particles hit the target in Example 5.7.8, but we could never see more than n failures in the setup of Theorem 5.7.11, no matter how long we observe.

A statistical model consists of an identification of random variables of interest (both observable and only hypothetically observable),a specification of a joint distribution for the observable random variables, the identification of any parameter of those distribution that are assumed unknown and possibly hypothetically observable, and a specification for a joint distribution for a unknown parameters.

From the given information and the definition of statistical model

The random variables of interest are the observables\({Z_1},{Z_2}...\)the times at which successive particles hit the target, and 尾, the hypothetically observable (parameter) rate of the Poisson process. The hit times occur according to a Poisson process with rate 尾 conditional on 尾.

Other random variables of interest are the observable inter-arrival times \({Y_1} = {Z_1}\) , and \({Y_k} = {Z_k} - {Z_k} - 1\)for. \(k \ge 2\)