Ex4soln

# Ex4soln - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1306: INTRODUCTORY STATISTICS Sem 1 2008/2009 EXAMPLE CLASS 4 ANSWERS Q1. A die is rolled a number of independent times until a six appears on the upside. (a) Let Y=the number of times(rolls of the die) that it takes until the first ”success”(a six appears) Y Geometric ( p = 1 6 ) (b) The p.d.f. is: P ( Y = y ) = q y - 1 p for y = 1 , 2 , 3 , ....... where q = 1 - p Thus, for this question f ( y ) = P ( Y = y ) = ( 5 6 ) y - 1 . ( 1 6 ) for y = 1 , 2 , 3 .... (c) Now X y =1 f ( y ) = X y =1 ( 5 6 ) y - 1 . ( 1 6 ) = ( 1 6 )(1 + 5 6 + 5 6 2 + ............ ) Recall: for a Geometric Progression S = a 1 - r where a = 1 st term and r = common ratio Therefore X y =1 f ( y ) = ( 1 6 )( 1 1 - 5 6 ) = ( 1 6 ) . 6 = 1 1

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(d) Now P ( Y = 1 , 3 , 5 , 7 , ...... ) = ( 1 6 )(1 + 5 6 2 + 5 6 4 + ............ ) since rolls of die are independent. P ( Y = 1 , 3 , 5 , 7 , ...... ) = X k =0 ( 5 6 ) 2 k . ( 1 6 ) = ( 1 6 )( 1 1 - ( 5 6 ) 2 ) = ( 1 6 ) . ( 36 11 ) = 6 11 (e) The distribution function F ( Y ) = P ( Y y ) = [ y ] X i =1 ( 5 6 ) i - 1 . ( 1 6 ) = ( 1 6 ) . (1 - 5 6 [ y ] ) (1 - 5 6 ) F Y ( y ) = 1 - 5 6 [ y ] for y > 0 0 y 0 2
Q2. (a)The moment generating function (m.g.f.) of a discrete random variable X is defined as: M X ( t ) = E ( e tX ) = X

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