Science:Math Exam Resources/Courses/MATH103/April 2014/Question 01 (b) ii
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Question 01 (b) ii 

Let be a probability density function defined on the interval . Determine whether the following statement is true or false: . 
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 

What does the quantity represent? 
Hint 2 

Start with . What can you say about in this case? 
Hint 3 

Conclude using . 
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 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. The quantity is the mean of a random quantity with probability density . Since is a density on the interval , the random quantity can take values in . Thus, the mean is also a value in . The statement is true. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. Multiplying the inequality by yields . Integrating this gives
The statement is true. 