Science:Math Exam Resources/Courses/MATH103/April 2014/Question 03 (b)
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Question 03 (b) 

Vancouver has some of the best salmon hatcheries (i. e. salmon farms) in the world. The probability that a young salmon takes days to swim from the Capilano Salmon Hatcher to the Pacific Ocean is given by the probability density function for . [It is assumed that it takes at least days for a salmon to reach the ocean.] Find the average (i. e. mean) time it takes for a young salmon to swim from the hatchery to the Pacific Ocean. Work must be shown for full marks. Simplify fully. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

The mean is given by

Hint 2 

Recall that, by definition, . 
Hint 3 

Try integrating by parts. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The mean is given by
Let and . Then and . Thus, by integration by parts,
It follows that the mean is
