Edit: Hey guys! Just solved a good chunk of the problems I was having with this, but still need some help with (c) if anyone can give me any pointers? Thanks!
A computer server crashes randomly. Time online without crashing can be estimated by an exponential distribution with expected value of 0.5yrs.
(a) What is the expected number of crashes in a 5 year time period?
$$E(X)=\frac{1}{\lambda}, \lambda=2, E(N(5))=\lambda t=2*5=10$$
(b) What is the probability that the server is online for at least a year without crashing?
$$P(t\ge1)=exp(-2\lambda)=exp(-2*1)=0.135$$
Figured out where I went wrong - Using $\lambda=0.5$ per year when that's the rate, the real value is $\lambda=2$ per year. This makes the above two equations equivalent once you plug the correct value back into it
(c) What is the probability that the server crashes within a year, given that it has been online for 3 months without crashing
$$P(N(1)\ge1\vert N(\frac{3}{12}=0)$$
I'm not sure where to go from here, so any help is appreciated. Thanks in advance!
For part (c), the key is the memoryless property of the exponential distribution. By this property, the fact it has been online for 3 months is irrelevant and thus the answer is simply the probability that the server crashes within 9 months.
You can read more on this here on page 2 and 3 : http://pages.cs.wisc.edu/~dsmyers/cs547/lecture_9_memoryless_property.pdf
It derives a mathematical proof for the memoryless property and also explains the intuition well.