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471fall_II - Prelim II 50 minutes SHOW ALL WORK(1[30 pts...

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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 0 1 2 π e - x 2 / 2 dx = 1 2 2 π e - y 1 / 2 / 2 y - 3 / 4 . (2) [30 pts] Suppose X and Y are independent exponential random variables with parameter λ . (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 λe - λx and λe - λy which is just λ 2 e - λ ( 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 X + Y .
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