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Unformatted text preview: Prelim II, 50 minutes SHOW ALL WORK! (1) [30 pts] Consider the Normal(0,1) random variable X . Find the density of the random variable Y = X 4 . f Y ( y ) = d dy P ( X 4 < y ) = d dy P (- y 1 / 4 < X < y 1 / 4 ) = d dy 2 Z y 1 / 4 1 2 e- x 2 / 2 dx = 1 2 2 e- y 1 / 2 / 2 y- 3 / 4 . (2) [30 pts] Suppose X and Y are independent exponential random variables with parameter . (a) What is the joint density, f X,Y ( x,y ), of the random vector ( X,Y )? Due to independence, it is the product of e- x and e- y which is just 2 e- ( x + y ) . (b) We know that one way to get the distribution of X + Y is take the convolution of the respective densities. Another way is to calculate P ( X + Y < z ) by considering a double integral of the form Z Z f X,Y ( x,y ) dxdy over a certain region. Indicate this region with a drawing and find the limits of integration which make the above integral equal to P ( X + Y < z ) (you do not need to calculate the integral). This is the distribution function of the sum...
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