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SampleFinalExamQuestionsStat3345Fall2010

SampleFinalExamQuestionsStat3345Fall2010 - SOME SAMPLE...

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SOME SAMPLE EXAMS PROBLEMS Problem 1 Suppose the X and Y are jointly distributed according to the probability density function has the following form f ( x, y ) = ( K x 2 + xy 2 · , if 0 < x < 1 and 0 < y < 2 , 0 , otherwise . (a) Find the constant K . (b) Find the marginal density function of X and Y . (c) Find P ( Y > 1 | X > 1 2 ) . (d) Find P ( Y > X ). (f) Find Var( X ) and Var( Y ) (g) Find Cov( X, Y ). (h) Find Var( X + Y ). (i) Find the conditional density function of Y given X = x . Problem 2 Given here is the joint probability mass function associated with data obtained in a study of automobile accidents in which a child (under age 5 years) was in the car and at least one fatality occurred. Specifically, the study focused on whether or not the child survived and what type of seatbelt (if any) he or she used. Define Y 1 = ( 0 , if the child survived, 1 , if not, and Y 2 = 0 , if no belt used, 1 , if adult belt used, 2 , if car-seat belt used. 1
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Notice that Y 1 is the number of fatalities per child and, since children’s car seats usually utilize two belts, Y 2
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