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assignment6 - Assignment 6 1 Let X and Y have joint pdf...

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Assignment6 1. Let X and Y have joint pdf: f X,Y ( x, y ) = k ( x + y ) for 0 x 1 , 0 y 1 . (a) Find k . (b) Find the joint cdf of (X,Y). (c) Find the marginal pdf of X and of Y . (d) Find P [ X < Y ], P [ Y < X 2 ], P [ X + Y > 0 . 5]. 2. The random vector ( X, Y ) is uniformly distributed (i.e., f ( x, y ) = k ) in the regions shown in the following figures and zero elsewhere. 1 1 ( i ) 1 1 ( ii ) 1 1 ( iii ) (a) Find the value of k in each case. (b) Find the marginal pdf for X and for Y in each case. (c) Find P [ X > 0 , Y > 0]. 3. Let X and Y be independent random variable. Find the expression for the proba- bility of the following events in terms of F X ( x ) and F Y ( y ). 1
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4. Let X and Y be the jointly Gaussian random variables with the means m 1 and m 2 and variances σ 1 and σ 2 respectively. The pdf is as following: f X,Y ( x, y ) = exp braceleftBigg - 1 2(1 - ρ 2 X,Y ) bracketleftbig( x - m 1 σ 1 ) 2 - 2 ρ X,Y ( x - m 1 σ 1 )( y - m 2 σ 2 ) + ( y - m 2 σ 2 ) 2 bracketrightbig bracerightBigg 2 πσ 1 σ 2 radicalBig 1 - ρ 2 X,Y
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