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# Joint Distribution

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Tim has two more problems to solve on his test, but he doesn't know how long it will take to finish them. He assigns mutually irrelevant normal distributions to the times, assigning a mean of 35 minutes to the first problem and 30 minutes to the second problem. He assigns a 95% chance that the first problem will take 35+-15 minutes(i.e. between 20 and 50 minutes), and a 95% chance that the second problem will take 30+-20 minutes.

Which interval below contains 95% of the probability of the time it will take Tim to finish both problems?

a) 65 +- 17.5 minutes
b) 65 +- 25 minutes
c) 65 +- 30 minutes
d) 65 +- 35 minutes

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#### Solution Preview

Solution. Assume that X and Y denote the time he will finsh the first problem and the second problem respectively. By the hypothesis we know that X~N(35, a^2), and Y~N(30, b^2). Since X and Y are irrelevant, equivalently ...

#### Solution Summary

Tim has two more problems to solve on his test, but he doesn't know how long it will take to finish them. He assigns mutually irrelevant normal distributions to the times, assigning a mean of 35 minutes to the first problem and 30 minutes to the second problem. He assigns a 95% chance that the first problem will take 35+-15 minutes(i.e. between 20 and 50 minutes), and a 95% chance that the second problem will take 30+-20 minutes.

Which interval below contains 95% of the probability of the time it will take Tim to finish both problems?

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