# Exponential Distribution, Poisson process

I have a question relating to Poisson which I would like to get some advice on. I am doing this on a self-help exercise, but unfortunately it doesn't provide answers for me to verify.

Basically in the question, it provides me with the premise that a server in a datacenter will under-go an unplanned system reboot every 40 days.

So, to find the probability that the time between TWO unplanned server reboots is less than 4 weeks, I use the expression $P( 2x < 28)$, where $2x$ is because of 2 unplanned server reboots and 28 because 4 weeks is equals to 28 days. I was wondering if I am on the right track on this?

## 1 Answer

Since the OP figured it out, I've written up a full outline of the answer now.

Step 1: Define your random variables.

Let $X$ be the time in days between two unplanned reboots. Because the reboots occur according to a Poisson process with rate 1 per 40 days, $X\sim\text{Exp}(\lambda)$, for $\lambda=1/40$

(That's if we're parameterizing exponentials in rate form ($f(x;\lambda) = \lambda e^{-\lambda x};\,x>0,\lambda>0$) rather than scale form ($f(x;\mu) = \frac{1}{\mu} e^{-x/\mu};\,x>0,\mu>0$).)

Step 2: Write the desired quantity in terms of $X$:

$P(X<28)=1-\exp(-28/40)\approx 0.5034$

• Hi Glen, I've updated the question. Many thanks for your guidance. What I am looking for is "the probability that the time between two reboots is <4 weeks" – user1275515 Feb 18 '14 at 4:39
• See my edited answer – Glen_b Feb 18 '14 at 6:07
• Thanks Glen (@glen_b), but since there are 2 unplanned server reboots, should it be 2x instead of x? – user1275515 Feb 18 '14 at 6:43
• The distribution of the time between events in a Poisson process is exponential with mean equal to <something I'll let you figure out>. What is asked for is the time between reboots. So no, everything is as it should be. – Glen_b Feb 18 '14 at 7:00
• Am I right to say that lamda is 40? – user1275515 Feb 18 '14 at 8:11