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

Consider the Normal (or Gaussian) probability density function (pdf ) given by for . (d) Find the variance . (Hint: what is the antiderivative of ?) 
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 

Recall that if is the probability density function on the domain , then the variance is given by

Hint 2 

To calculate the integral, use integration by parts. 
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. Using Hint and part (a), we have
Note that the integrand is an even function, we have
To calculate the integral, we first check the antiderivative of . By the change of variable , we have and hence
Then, using the integration by parts with and . we have
Since is an even function and from part (c), we have
