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Unformatted text preview: STA 4321/5325 - Spring 2010 Exam 3 March 29, 2010 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. GOOD LUCK. 1 Problem 1: The probability density function of a continuous random variable X is f ( x ) = braceleftbigg 2(1- x ) , for 0 x 1 , otherwise (a) [4 pts] Find the distribution function F ( x ) of X . (Be sure to account for all values of x ). Solution: For 0 x 1, F ( x ) = integraldisplay - f ( t ) dt = integraldisplay x 2(1- t ) dt = 2 bracketleftbigg t- 1 2 t 2 bracketrightbigg x = 2( x- 1 2 x 2- 0) = 2 x- x 2 Clearly, for x &lt; 0, F ( x ) = 0, and (since f ( x ) integrates to 1) for x &gt; 1, F ( x ) = 1, so F ( x ) = , for x &lt; 2 x- x 2 , for 0 x &lt; 1 1 , for x 1 (b) [3 pts] Compute P ( X &gt; . 5). P ( X &gt; . 5) = integraldisplay . 5 f ( x ) dx = integraldisplay . 5 2(1- x ) dx = 2 bracketleftbigg x- 1 2 x 2 bracketrightbigg 1 . 5 = 2 parenleftbigg 1- 1 2 1 2- (0 . 5- 1 2 . 5 2 ) parenrightbigg = 0 . 25 (c) [3 pts] Find the expected value of X ....
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- Spring '08