# 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 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?

##### 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 ...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

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