Science:Math Exam Resources/Courses/MATH103/April 2016/Question 09 (d)
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Question 09 (d) 

Let be a continuous random variable taking values in with probability density function , mean value , variance , median value , and cumulative function . Show that . 
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 

Apply integration by parts to the answer in part (a). 
Hint 2 

Remember that for a probability density function and the cumulative density function is given by 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. By part (a), we have Following the hint, we integrate by parts. Recalling the integration by parts formula, we take and , so that (i.e., is an antiderivative of ; see hint 2). Then Now but is a probability density function, so Therefore 