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# hw17 - H s What is the probability that exactly four H s...

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ENEE 324 ASSIGNMENT 17 Due Tue 11/10 Problem 17A The density of X is given by f X ( x ) = ( 2 x, 0 < x < 1; 0 , otherwise. The conditional density of Y given X = x is given by f Y | X ( y | X = x ) = ( cx 2 /y 3 , y x ; 0 , otherwise. (i) (2 pts.) Shade the region on the first quadrant over which the joint density f XY ( x, y ) is nonzero, and give an equation for it (the joint density). (ii) (2 pts.) Determine the value of c . (This can be done using the expression for f Y | X ( · | · ) only.) (iii) (6 pts.) Determine and sketch f Y ( y ), distinguishing between the cases y 1 and y > 1. (iv) (2 pts.) Determine and sketch f X | Y ( x | Y = 1 / 2). (v) (2 pts.) Determine and sketch f X | Y ( x | Y = 2). (vi) (6 pts.) Compute the following expectations: E [ Y | X = x ], E [ X 2 | Y = 1 / 2] and E [ XY ]. Problem 17B (i) (7 pts.) A coin drawn at random from a large collection of bent coins has probability of H equal to X , where f X ( x ) = 12 x 2 (1 - x ) x (0 , 1) The coin is tossed six times. What is the expected number of
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Unformatted text preview: H s? What is the probability that exactly four H s are obtained? (ii) (6 pts.) Suppose that the number N of homework problems assigned during a semester is a random variable with geometric pmf P [ N = n ] = 2-n , n ≥ 1 Given N = n , the total time T (e.g., in hours) spent on homework has conditional pdf f ( t ) = 3 n t n-1 ( n-1)! · e-3 t , t ≥ Determine E [ T | N = n ] (a known integral, already encountered in this class) and hence E [ T ]. (iii) (7 pts.) Let X be Gaussian with E [ X ] =-2 and Var[ X ] = 9. Let the conditional distribution of Y given X = x also be Gaussian with mean E [ Y | X = x ] =-2 x + 1 and variance E [ Y 2 | X = x ]-( E [ Y | X = x ]) 2 = 4 Determine E [ Y ], Var[ Y ] and E [ XY ]....
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