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# ps7 - (b Are X and Y statistically independent Justify your...

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ECE 804, Random Signal Analysis Nov. 16, 2009 OSU, Autumn 2009 Due: Nov. 23, 2009 Problem Set 7 Problem 1 Random variables X and Y have a joint density, f X,Y ( x, y ) = k y inside the triangular region shown below and 0 elsewhere. -2 -1 0 1 2 -1 -0.5 0 0.5 1 1.5 2 X Y In what follows you may or may not need the following: R 1 0 ln x dx = - 1. (a) Find k . (b) Find and plot the marginal densities f X ( x ) and f Y ( y ). (c) Which of the following statements is correct? Explain. (i) X and Y are independent. (ii) X and Y are uncorrelated. (d) Let Z = XY . Find f Y,Z ( y, z ) Problem 2 Let Θ and R be independent random variables, where Θ is uniformly distributed on [0 , π 2 ], and f R ( r ) = λe - λr u ( r ). Let X = R cos(Θ) and Y = R sin(Θ). (a) Find the joint pdf of X and Y . Be sure to clearly state the region in the ( x, y ) plane in which this joint pdf is nonzero.

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Unformatted text preview: (b) Are X and Y statistically independent? Justify your answer. 1 Problem 3 At a party n people put their hats in the center of a room, where the hats are mixed together. Each person then randomly selects one with equal probability. Let us deﬁne ~ X = [ X 1 X 2 ··· X n ], where X i = ( 1 , if the i th person selects his or her own hat , otherwise. (a) Find E [ X i ] and var( X i ). (b) Find the covariance matrix, C ~ X . For n = 3, ﬁnd its eigenvalues and eigenvectors. Write it in the form C ~ X = V Λ V T , where Λ is a diagonal matrix. (c) Let S = X 1 + X 2 + ··· + X n . Find E [ S ] and σ 2 S . 2...
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ps7 - (b Are X and Y statistically independent Justify your...

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