Midterm2 - X . 2 3. [20 pts.] Calculate the PDF...

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Stat 5101 Second Midterm Exam November 18, 2009 Name Student ID The exam is closed book and closed notes. You may use one 8 1 2 × 11 sheet of paper with formulas, etc. You may also use the handouts on “brand name distributions” and Greek letters. Put all of your work on this test form (use the back if necessary). Show your work or give an explanation of your answer. No credit for numbers with no indication of where they came from. The points for the questions total to 100. There are 5 pages and 5 prob- lems. 1. [20 pts.] Suppose X is a random variable having probability density func- tion (PDF) given by f ( x ) = 6 5 ( x + x 2 ) , 0 < x < 1 . Find the PDF of the random variable Y = X 1 / 2 . The definition of a function describes the domain as well as the rule. 1
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2. [20 pts.] Suppose X is a random variable having PDF given by f ( x ) = 2 x 3 , 1 < x < . (a) Find the mean of the distribution of X . (b) Find the median of the distribution of
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Unformatted text preview: X . 2 3. [20 pts.] Calculate the PDF corresponding to the DF F ( x ) = , x x 2 / 2 , x 1 1-(2-x ) 2 / 2 , 1 x 2 1 , 2 x 3 4. [20 pts.] Suppose the random vector ( X,Y ) has the PDF f ( x,y ) = 4(1 + xy ) / 5 , &lt; x &lt; 1 , &lt; y &lt; 1 . Find the conditional expectation of Y given X . 4 5. [20 pts.] Suppose the conditional distribution of Y given X is NegBin( r,X ) where r is a known positive integer, and suppose the marginal distribution of X is Beta( 1 , 2 ). What is the conditional distribution of X given Y ? Since this is a brand name distribution, no integrals need be done, it is enough to name the distribution and give its parameters as a function of Y , r , 1 , and 2 . 5...
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Midterm2 - X . 2 3. [20 pts.] Calculate the PDF...

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