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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306 INTRODUCTORY STATISTICS (SEMESTER 1 2008/2009) Assignment 2 Hand in solutions by 22th October, 2008 1. Suppose a continuous random variable Y whose pdf is given by f Y ( y ) = & 1, for < y < 1; 0, elsewhere. (a) Find its MGF. (b) Find E ( Y ) ; E ( Y 2 ) and V ar ( Y ) . 2. Suppose the MGF of a random variable Y is M Y ( t ) = e 3 t +8 t 2 . (a) Find the MGF of a random variable Z = 1 4 ( Y & 3) . (b) Find the mean and the variance of Z . (a) Show that the conditional distribution function of the continuous random variable Y , given a < Y ¡ b , is given by F Y ( y j a < Y ¡ b ) = 8 > > < > > : for y ¡ a F Y ( y ) & F Y ( a ) F Y ( b ) & F Y ( a ) for a < y ¡ b 1 for y > b (b) Di/erentiate the result of part (a) to obtain the conditional pdf of Y , given a < Y ¡ b and show that the expectation u ( Y ) conditional on a < Y ¡ b is E ( u ( Y ) j a < Y ¡ b ) = Z b a u ( y ) f Y ( y ) dy Z b a f Y ( y )...
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- Fall '08