Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top
Jobs are submitted for a computer according to a Poisson process with a rate λ
(jobs / hour).

Determine the probability that no job is lost if the computer crashes in the:
in the first 'a' minutes 
in the last 'b' minutes of one hour.

(a job is lost if the computer is not working at the moment it is submitted).

I tried to modify λ to match the 'a' and the first few minutes 'b' past, but ended up losing. But I could see a few things: The problem whether the probability of:

1) No job lost if the computer crashes in the first 'a' minutes;

2) No job lost if the computer crashes in the last 'b' minutes;

Really, I'm confused, can someone help me?

share|cite|improve this question
up vote 2 down vote accepted


Let's define $X$ to be the number of jobs submitted to the computer in 1 hour. $X$ is Poisson with mean $\lambda$.

Because of the memoryless property of a Poisson event, it doesn't matter which $a$ minutes timeframe was taken to measure. First, convert $a$ mins to hours like so:

$$a \text{ mins} = \frac{a}{60} \text{hours}$$

Number of jobs lost in $a$ minutes will follow a Poisson distribution, let's call it $A$ with its parameter, let's call it $\lambda_A$ to be:

$$\begin{align*} \lambda_A &=\frac{a}{60} \times \lambda \\&= \frac{a\lambda}{60} \end{align*} $$

Plug this into the formula for Poisson for $P(A=0)$ and you will be fine.


This is a little bit trickier. Let's call this parameter $\lambda_B$. This time period can be defined as

$$ \lambda_B = \frac{60-b}{60} \text{ hours} $$

Do the same thing and you should get your correct answer in terms of $\lambda$.

I'll give you another hint - the probability that there are no jobs received in a 1 hour period is

$$P(X=0) = \frac{\lambda^0e^{-\lambda}}{0!}$$

Recall that this $X$ is actually the $X$ referred to at the beginning of my answer.

share|cite|improve this answer
Thank you! I was thinking that solution maybe diferent because this problem has two time periods: first 'a' minutes and last 'b' minutes. This solution seems like that one that I write for another problem.…. Thank you, you rocks!! – Richard Feb 17 '13 at 11:51
If I multiply the result, will I find the intersection? the probability that no job is lost in the first 'a' minutes AND in the last 'b' minutes? – Richard Feb 17 '13 at 11:54

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.