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STAT272 PROBABILITY Assignment 2, 2010 Due date and time: Friday 14 May (Week 10), in your tutorial. Question 1 The random variable X is distributed...

Question 4
The random variable X is exponentially distributed with mean : Let Y be the integer part of
X: That is, if X 2 [n, n + 1) ; then Y = n. Find the pgf of Y and hence E (Y ) ,E fY (Y ô€€€ 1)g and var Y:
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STAT272 PROBABILITY Assignment 2, 2010 Due date and time: Friday 14 May (Week 10), in your tutorial. You are required to use the Statistics Department&s assignment cover sheet, which is available on Blackboard. Question 1 The random variable X is distributed exponentially with mean 1 . Find the pdf of (a) Y = log X ; (b) Y = e X ; (c) Y = ( X & 1) 2 ; Question 2 The random variable X has the standard Cauchy distribution : The random variable Y = X 2 : Find the pdf of Y: Question 3 The random variable X is distributed normally with mean 0 and variance 1 : (a) Find directly the moment generating function of Y = X 2 (do not &nd the pdf of Y ); (b) Calculate the moment generating function of the random variable Z which has pdf f Z ( z ) = ( e & z= 2 p 2 &z ; z > 0 0 ; z ± 0 : (c) What do you conclude from the above two results? Question 4 The random variable X is exponentially distributed with mean &: Let Y be the integer part of X: That is, if X 2 [ n;n + 1) ; then Y = n: Find the pgf of Y and hence E ( Y ) ;E f Y ( Y & 1) g and var Y: 1
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Question 5 The random variables X 1 and X 2 are i.i.d, with probability functions f ( x ) = & (1 & p ) x & 1 p ; x = 1 ; 2 ;::: 0 ; otherwise. The random variable Y is de&ned as the minimum of X 1 and X 2 : Find the probability generating function of Y: Hint: Ez Y = 1 X x 1 =1 1 X x 2 =1 z min( x 1 ;x 2 ) f ( x 1 ) f ( x 2 ) : Question 6 The random variables X 1 ;:::X n are i.i.d. N (0 ; 1) random variables. The sample variance is de&ned as S 2 = 1 n & 1 n X i =1 ± X i & X ² 2 : In STAT271 we shall show that ( n & 1) S 2 has the & 2 distribution with degrees of freedom n & 1 : For the special case where n = 2 ; show this directly. You will need to specify every rule you have used. 2
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