Problem 2, August 2004

In this problem we will consider a certain electrical component
where errors of to types may occur. Let us call these error
types A and B, respectively. We make the following assumptions:

- The component is replaced immediately when an error occurs.

- Errors of type A occur as a Poisson process of intensity
rho1.

- Errors of type B occur as a Poisson process of intensity
rho2.

- The two processes are independent.

(a) Explain that errors of the component occur as a Poisson
process of intensity rho1+rho2. Give a (practical) natural
objection to modelling errors of the component as a Poisson
process of constant rate.

(b) Given that an error has occurred once in the time interval
(0,t], what is the probability that the error was of type A?

(c) Given that n errors have occurred in (0,t], derive the
distribution of the number of errors in the time interval
(0,u], where u<=t. Find the probability that exactly k (<=n)
errors occurred in the interval (u,t] when you know that n
errors in total occurred in (0,t].