sampleexam3solns - STA 4321/5325 Spring 2010 Sample Exam 3...

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STA 4321/5325 - Spring 2010 Sample Exam 3 Note: This is an example adapted from previous STA 4321/5325 exams. It is intended to be of approximately the same length and style as the actual exam. However, it is NOT guaranteed to match the content or coverage of the actual exam. DO NOT use this as your primary study tool! Full Name: KEY On my honor, I have neither given nor received unauthorized aid on this examination. Signature: This is a 50 minute exam. There are 4 problems, worth a total of 40 points. You may use one letter-size sheet of your own notes and a standard scientific calculator. You may not use any other references (e.g. books, notes, or text-capable devices). You are not required to bring a calculator – you may leave your answers in an arithmetic form from which the numerical answer could be immediately calculated. Remember to show your work. Answers lacking adequate justification may not receive full credit. Write all answers in the spaces provided. If you require more space to write your answer, you may use the back side of the page. Please have your UF student ID card available. 1
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Problem 1: Let X has a density function f ( x ) = braceleftbigg x + cx 2 , for 0 x 2 0 , elsewhere (a) [2 pts] Find the value of c. Solution: We know that integraltext -∞ f ( x ) dx = 1, then 1 = integraldisplay 2 0 ( x + cx 2 ) dx = parenleftbigg x 2 2 + c x 3 3 parenrightbigg 2 0 = parenleftbigg 2 2 2 + c 2 3 3 parenrightbigg - parenleftbigg 0 2 2 + c 0 3 3 parenrightbigg = 2 + c 8 3 Hence, c = - 3 8 and f ( x ) = braceleftbigg x - 3 8 x 2 , for 0
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