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tutorial8 - X(d Calculate P X> Y(e Find the conditional...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 8 EX.1 Consider the following joint pmf of X and Y . Y X 0 1 2 0 0 . 1 0 . 3 0 . 05 1 0 . 2 0 . 25 0 . 1 Find the values of (a) P ( X = Y ) and P ( X > Y ); (b) the marginal pmfs of X and Y ; (c) the pmf of X + Y ; (d) E ( X ) ,E ( Y ) ,V ar ( X ) and V ar ( Y ); (e) Cov ( X,Y ) and Corr ( X,Y ); then show that X and Y are NOT independent; (f) the conditional pmf of Y given X = 1. 1

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EX.2 The joint probability density function of X and Y is given by f ( x,y ) = ( c ( x 2 + xy 2 ) , if 0 < x < 1, 0 < y < 2; 0 , otherwise. (a) Are X and Y independent? (b) Find the value of c . (c) Find the marginal probability density function of
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Unformatted text preview: X . (d) Calculate P ( X > Y ). (e) Find the conditional probability density function of Y given X = x . (f) Calculate P ( Y < 1 2 | Y > X ). 2 EX.3 Show that (a) V ar ( A + B ) = V ar ( A ) + V ar ( B ) + 2 Cov ( A,B ); (b) V ar ( kX ) = k 2 V ar ( X ); (c) Cov ( aX,bY ) = abCov ( X,Y ); (d) ρ XY ≥ -1 by using the fact that V ar ( X σ X + Y σ Y ) ≥ 0; (e) | ρ XY | ≤ 1 by using the above results and the fact that V ar ( X σ X-Y σ Y ) ≥ 0. 3...
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tutorial8 - X(d Calculate P X> Y(e Find the conditional...

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