# Conditional probabilities involving the exponential distribution

The number of years the laptop functions is exponentially distributed with mean = 5 years. If a customer purchased an old laptop which was used for last two years, what is the probability that it will function for at least 3 years?

The probability you want is $P(X>5|X>2)$ where $X$ has an exponential distribution. We're seeing whether it's greater than 5 because you want to know if it lasts 3 additional years, after it's already been functioning for 2 years.

Using the CDF of the exponential distribution, with $\lambda$ referring to the "rate parameter",

$$P(X \leq 5 | X> 2 ) = \frac{ P( X \leq 5 \ \cap \ X > 2) }{P(X>2)} = \frac{ P( 2 < X \leq 5 ) }{P(X>2)} = \frac{ e^{-2\lambda} - e^{-5 \lambda} }{e^{-2 \lambda} } = 1 - e^{-3\lambda}$$

This means $$P(X>5|X>2) = 1- P(X \leq 5 | X> 2 ) = e^{-3 \lambda}$$

In your case the mean is 5, which means $\lambda = 1/5$ using the parameterization I've used here, so the probability you want is $e^{-3/5} \approx .549$.

Note that this is the same as the probability that a brand new computer lasts for 3 years. This is because the exponential distribution has something called the memoryless property.

I hope this helps!!

• The exponential distribution has a closed form cdf. So it is very easy to do calculations like this. Well done anyway +1 – Michael R. Chernick Sep 20 '12 at 0:58

Based on the lack of memory property for the exponential distribution, the remaining life has the same distribution as it would have if the laptop were brand new. So just integrate the given exponential density from 3 to infinity to obtain the desired probability since P[X>5|X>2]=P[X>3].

• Good answer Michael Chernick! For your information, you can prove the memoryless property by using the definition of conditional probability and the form the CDF of the exponential distribution. If you are interested in this and are not familiar with these topics (which you may not be exposed to until a college statistics class) then you can consult the wikipedia pages about these topics or see my answer. I hope this helps!! – user41583 Sep 19 '12 at 18:22
• Thanks but I have a PhD in statistics and am very familiar with the exponential distribution. – Michael R. Chernick Sep 19 '12 at 19:04
• Glad I could help! Do let me know if you have any further confusion about the exponential distribution or conditional probability. I have a pretty good understanding of these topics! – user41583 Sep 19 '12 at 19:10
• Hi Michael Chernick! I'm new to this site and often forget to use the '@' symbol :-) My comments were addressed to you since you seem to have some confusion about conditional probability and the exponential distribution. After you've looked at the wikipedia pages, let me know if you have any further questions. I may be able to help! – user41583 Sep 19 '12 at 21:44
• Hi @Michael! I didn't mean to convey sarcasm or any kind of disrespect. I only thought that you may have refrained from showing that $P(Y>5|Y>2) = P(Y>3)$ because you weren't sure how to sure how to proceed with the proof. It's possible that you haven't covered that yet in your PhD studies. When you do, you can always "@" me here if you have any questions! – user41583 Sep 19 '12 at 23:58