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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!

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

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