Science:Math Exam Resources/Courses/MATH101/April 2009/Question 05
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Question 05 

Let X be a random variable with probability density function: (a) Find the value of k. (b) Find the mean . (c) Find an algebraic equation satisfied by the median m. 
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 

For ƒ to be a probability density function it needs to satisfy two conditions:
Choose k such that both conditions are fulfilled. 
Hint 2 

The formula for the mean is given by 
Hint 3 

The median m is that number which satisfies 
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. (a) Using hint 1 we see that k needs to be a positive constant. We can calculate k directly by integrating: The only way that this integral has value 1 is when k = 3. 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. (b) Using the second hint and the fact that k = 3 the calculation is straight forward

Solution 3 

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Please rate my easiness! It's quick and helps everyone guide their studies. (c) Following the third hint we need to find that value m which satisfies Since it is clear that m needs to be a number between 0 and 1. Therefore the algebraic equation that m satisfies is the following; 