Test 4 Solutions

Test 4 Solutions - STAT 230 Test 4 June 28, 2006 3:30 4:10...

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Unformatted text preview: STAT 230 Test 4 June 28, 2006 3:30 4:10 pm 1. Let f X ( x ) = P ( X = x ) = pq x , x = 0 , 1 , 2 , and f Y ( y ) = P ( Y = y ) = pq y , y = 0 , 1 , 2 , , where q = 1- p , X and Y are independent. [2] (a) Show that the moment generating function of X is M ( t ) = p/ (1- qe t ). Soln : M ( t ) = E ( e tX ) = i =0 e tx pq x = p i =0 ( e t q ) x = p/ (1- qe t ) for t such that pe t < 1. [2] (b) Use any suitable method to find E ( X ). Soln 1 : E ( X ) = i =0 xpq x = pq/ (1- q ) 2 = q/p . Soln 2 : M ( t ) = pqe t / (1- qe t ) 2 , E ( X ) = M (0) = q/p . [2] (c) Let T = X + Y . Find the probability function of T . Soln : f T ( t ) = P ( X + Y = t ) = t i =0 P ( X = i,Y = t- i ) = t i =0 ( pq i )( pq t- i ) = ( t + 1) p 2 q t , t = 0 , 1 , 2 , . [2] (d) Let T = X + Y . Find the conditional distribution of X given T = n . Soln : f X | T ( x | n ) = P ( X = x | T = n ) = P ( X = x,T = n ) /P ( T = n ) = P ( X = x,Y = n- x ) /P ( T = n ) = ( pq x )( pq n- x ) / { ( n + 1) p 2 q n } = 1 / ( n + 1), x = 0 , 1 , ,n . The conditional distribution of X given T = n is discrete uniform over [0 ,n ]....
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Test 4 Solutions - STAT 230 Test 4 June 28, 2006 3:30 4:10...

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