I have a quick question which I can't figure out how to start. I actually do not understand how to model the probabilities. can anyone help? Thanks. Here it is:
A system will function as long as at least one of the three components functions. When all three components are functioning, the distribution of the life of each is exponential with parameter $\frac{\lambda}{3}$. When only two are functioning, the distribution of the life of each is exponential with parameter $\frac{\lambda}{2}$; and when only one is functioning, the distribution of life is exponential with parameter $\lambda$.
(a) What is the distribution of the lifetime of the system?
(b) Suppose now that only one component (of the three components) is used and it is replaced when it fails. What is the distribution of the lifetime of such a system?
With regards to modeling the probability distributions, do the phrases in the given text mean that: $$f_{X_1|X_2,X_3}(x_1|x_2,x_3)=f_{X_2|X_1,X_3}(x_2|x_1,x_3)=f_{X_3|X_1,X_2}(x_3|x_1,x_2)=\frac{\lambda}{3}e^{\frac{-\lambda x_i}{3}}$$ $$f_{X_1|X_2}(x_1|x_2)=f_{X_1|X_3}(x_1|x_3)=f_{X_2|X_1}(x_2|x_1)=f_{X_2|X_3}(x_2|x_3)=f_{X_3|X_1}(x_3|x_1)=f_{X_3|X_2}(x_3|x_2)=\frac{\lambda}{2}e^{\frac{-\lambda x_i}{2}}$$ and $$f_{X_1}(x_1)=f_{X_2}(x_2)=f_{X_3}(x_3)=\lambda e^{-\lambda x_i}$$
The point is that the failure rate of resistors is constant at $\lambda$ regardless of how many are operating. You have a Poisson process where in each time interval $dt$ the chance of a failure is $\lambda dt$ The system fails if there have been three or more events, so at time $t$ the average number of failures is $\lambda t$ and the chance of failure by time $t$ is $\sum_{k=3}^\infty \frac {(\lambda t)^3}{k!}e^{-\lambda t}$
As I read it, you now need four failures with the same failure rate as above. So change $3$ to $4$. If you keep replacing components, it never fails.