# Probability of computer server being online without crashing

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!

• whats the probability that it goes 9 months without crashing? – shai horowitz Jun 5 '18 at 23:09