test2 - chosen at random Find the cumulative distribution...

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MAT 521 Test 2 June 21, 2005 Instructions: Write the answers and show all your work in the blue books. There are 5 problems. Make sure you do all 5. No books, notes, or collaboration with others. Problem 1. (8 points) Let a joint probability density function be given by f ( x, y ) = c ( x - y ) , 0 < y < x < 1 . for some constant c . a. Find the constant c . b. Set up the iterated integrals needed to find P (2 Y < X ) . You need not evaluate the integrals. Problem 2. (6 points) Let a point be chosen at random from the triangular region having vertices at (0,0), (0,1), and (2,0). Find the probability density function of the y coordinate of the chosen point. Problem 3. (5 points) A stick one foot long is broken at a point that is
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Unformatted text preview: chosen at random. Find the cumulative distribution function of L , the longer of the two pieces that result. (Hint: Let X be the position of the break. Then X is uniformly distributed on [0,1].) Problem 4. (5 points) Let X have the bilateral exponential density: f ( x ) = 1 2 e-| x | ,-∞ < x < ∞ . Find the density of X 2 using the cumulative distribution function method. Problem 5. (6 points) Let X and Y be i.i.d, each having as density function the Cauchy density 1 π 1 1 + x 2 . Find the joint density function of U = 2 X + Y and V = Y. 1...
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