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Homework 6

# Homework 6 - D Is D one of the common random variables...

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FALL 2011 Instructor: Sujay Sanghavi Homework 6 Due: October 20th in class Problem 1 (a) A ﬁre station is to be located at a point a along a road of length A , 0 < A < . If ﬁres will occur at points uniformly chosen on (0 ,A ) , where should the station be located so as to minimize the expected distance from the ﬁre? That is, choose a so as to minimize the quantity E [ | X - a | ] when X is uniformly distributed over (0 ,A ) . (b) Now suppose that the road is of inﬁnite length–stretching from point 0 outward to . If the distance of a ﬁre from point 0 is exponentially distributed with rate λ , where should the ﬁre station now be located? That is, we want to minimize E [ | X - a | ] with respect to a when X is now an exponential random variable with parameter λ . Problem 2 A random point ( X,Y ) on a plane is chosen as follows: X and Y are chosen independently, with each one being a Gaussian random variable with zero mean and variance of 1. Let D be the square of the (random) distance of the point from the origin. Find the PDF of
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Unformatted text preview: D . Is D one of the common random variables? Problem 3 Suppose the length L and width W of a rectangle are independent and each uniformly distributed over the interval [0 , 1] . Let C = 2 L + 2 W (the length of the perimeter) and A = LW (the area). Find the means, variances, and probability densities of C and A . Problem 4 Consider two random variables X and Y . For simplicity, assume that they both have zero mean. (a) Show that X and E [ X | Y ] are positively correlated. (b) Show that Cov ( Y,E [ X | Y ]) = Cov ( X,Y ) . Problem 5 Let random variables X and Y be jointly uniformly distributed on the region { ≤ x ≤ 1 , ≤ y ≤ 1 } ∪ {-1 ≤ x < ,-1 ≤ y < } . (a) Determine the value of f XY on this region. (b) Find f X , the marginal PDF of X . (c) Find E [ Y | X = x ] for | x | ≤ 1 . (d) What is the correlation coefﬁcient of X and Y ? (e) Are X and Y independent? (f) What is the PDF of Z = X + Y ?...
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