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I have the following question i am working on :

  • Assume that the time it takes before a hard disk drive crashes is exponentially distributed with a mean of 5 years. Consider a RAID system consisting of 100 hard disk drives. Show that the probability that one of the disks crashes within a month is more than 80% and within a day more than 5%.

I am aware that i have to put what i have attempted but in reality i have made lots illogical attempts to solve the question but can't. I am just confused as to what lambda and t is I will appreciate an explanatory text of the different variables and how they will fit into the poisson formula

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  • $\begingroup$ @Tyler if the time between events is exponentially distributed, the number of events in a certain time period has poisson distribution. $\endgroup$
    – KSmarts
    Dec 23, 2014 at 14:21
  • $\begingroup$ The mean failure rate, $\lambda$, is 1 per 5 years, or $\frac{1\ \textrm{failure}}{5\ \textrm{years}}$. Convert this rate to "failures per month". Now use it in your Poisson distribution. $\endgroup$
    – Emily
    Dec 23, 2014 at 14:41
  • $\begingroup$ @Arkamis : You neglected the fact that there are 100 disks. $\endgroup$
    – Michael Hardy
    Mar 14, 2016 at 16:06

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The rate is $\lambda = 100\times \dfrac 1 {5\text{ years}} = 20\text{ per year} = \dfrac{20}{12} \text{ per month} = \dfrac 5 3 \text{ per month}$.

The probability that the number of occurrences in a month is $0$ is therefore $$ \frac{(5/3)^0 e^{-5/3}}{0!} \approx 0.1888756. $$

Proceed similarly for a period of one day.

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