Two downloads both done in two minutes

Question

Two files start downloading at the same time on independent connections. Download A's finish time is exponential with a 2-minute average. Download B's finish time is exponential with a 4-minute average.

What is the probability that both downloads finish within 2 minutes?

Accepted Answer

"Both finish within 2 minutes" is a statement about the slower download, the maximum. Since the two connections are independent, the joint chance is the product of the two individual finish-by-2-minutes chances. For an exponential finish with rate \lambda, the chance of finishing by time t is 1 - e^{-\lambda t}. Convert each mean to a rate (\lambda = 1/	ext{mean}): Download A: mean 2 min, so \lambdaA = 	frac{1}{2}, and \lambdaA t = 	frac{1}{2}\cdot 2 = 1. Chance by 2 min = 1 - e^{-1} \approx 0.6321. Download B: mean 4 min, so \lambdaB = 	frac{1}{4}, and \lambdaB t = 	frac{1}{4}\cdot 2 = …

Two downloads both done in two minutes

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