This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ]....
View Full
Document
 Fall '06
 various

Click to edit the document details