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?
[Math] Conditional probabilities involving the exponential distribution
probabilityprobability distributions
Best Answer
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!!