Unformatted text preview: Exercise 4 Let X and Y be a couple of random variables with pdf f ( x,y ) = 1 x 2 y 2 x ≥ 1 , y ≥ 1 a. (6 points) ﬁnd the joint pdf of U = XY and V = X/Y b. (8 points) ﬁnd the marginal pdf of U and V Exercise 5 (8 points) Let X have a beta distribution with parameters α and β ; the pdf of X is given by: f ( x ) = Γ( α + β ) Γ( α )Γ( β ) x α1 (1x ) β1 < x < 1 Show that E ( X ) = α α + β and V ar ( X ) = αβ ( α + β + 1)( α + β ) 2 Exercise 6 (8 points) Fifty numbers are rounded oﬀ to the nearest integer and then summed. If the individual roundoﬀ errors are uniformly distributed over the interval (1 / 2 , 1 / 2). Find the probability that the resultant sum diﬀers from the exact sum by more than 3. good luck...
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 Spring '10
 VariousTeachers
 Statistics, Probability, Standard Deviation, Probability theory, 800 hours, 100 hours, 10000 hours, AMERICAN UNIVERSITY OF BEIRUT

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