Science:Math Exam Resources/Courses/MATH101/April 2006/Question 06
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Question 06 

FullSolution Problems. Justify your answers and show all your work. Simplification of answers is not required. X is a random variable with probability density function where . Find the median of X. 
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 median m is defined as that number, which satisfies 
Hint 2 

A simple substitution may be helpful to evaluate the integral. 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Probability was not covered this year, hence this is not material for the April 2012 exam. Since f(x) = 0 for x≤ 0 it follows that m > 0. Now we can directly compute m: We use the substitution u = x^{2}, du = 2x dx. Then u(0) = 0 and u(m) = m^{2}. Hence In other words, m satisfies Note that we can rule out the solution since we already noted that m is positive. Hence, the solution is . 