Science:Math Exam Resources/Courses/MATH103/April 2015/Question 07 (b)
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Question 07 (b) 

Consider the Normal (or Gaussian) probability density function (pdf ) given by for . (b) Find the median 
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 given probability density function is an even function. 
Hint 2 

If is a probability density function, it 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 

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Please rate my easiness! It's quick and helps everyone guide their studies. By the definition of the median, we need to find such that
Since is pdf, by Hint 2, we have . On the other hand, we observe that is an even function, which implies that
Therefore, we get
and hence . In other words, the median is . 