Science:Math Exam Resources/Courses/MATH103/April 2010/Question 01 (f)
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Question 01 (f) 

Multiple Choice Question: Exactly ONE of the answers provided is correct. There is no partial credit in this questions. A continuous random variable has a probability density function of the form where and are constants. The values of and should be

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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Recall that a probability density function f(x) has to satisfy two conditions:

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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Using the hint, we first integrate f(x) and make sure it integrates to 1. Keep in mind that if : Note that it is no surprise that A cancels here, because the term Ax is odd an thus cancels when integrating from 1 to 1. The calculation above tells us that B = 3/2. The constant A needs to ensure that , for all x. For very small (in absolute value) negative numbers x, the linear term Ax dominates the quadratic term 3/2 x^{2}, so we suspect that A = 0 is the only option to keep f(x) positive. But let's do the math:
Hence x = A/3 is the local minimum of f(x). At this point the value of f is for any A. Since we need all points, including this critical point, to satisfy then we demand A=0. With this choice of A, notice that so we have that the local minimum is the global minimum and therefore that for all x. Final answer: iii. 