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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Recall that a probability density function f(x) has to satisfy two conditions:
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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 x2, 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.