Chapter 4 Continuous Random Variables Appendix (First and Second Moments of Uniform Random Variable)

Chapter 4 Continuous Random Variables Appendix (First and Second Moments of Uniform Random Variable)

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306 INTRODUCTORY STATISTICS (SEMESTER 1 2008/2009) Uniform Distribution By using the method of MGF, the mean and variance of a uniform random variable Y U [ a; b ] can be obtained as follows: Since we know the MGF of Y is M Y ( t ) = e tb ± e ta t ( b ± a ) ; M Y ( t ) w.r.t. is dM Y ( t ) dt = 1 b ± a be bt ± ae at t ± e bt ± e at t 2 ± = ( bt ± 1) e bt ± ( at ± 1) e at ( b ± a ) t 2 Clearly, when t = 0 , both numerator and denominator are zero. It would be better if the case t ! 0 is considered instead. Then, lim t ! 0 dM Y ( t ) dt = b 2 ( bt + 1) e bt ± a 2 ( at + 1) e at 2( b ± a ) ² ² ² ² t =0 = b 2 ± a 2 2( b ± a ) = ( b ± a )( b + a ) 2( b ± a ) = b + a 2 . where the L±Hopital±s rule is used for twice here. Analogously, the second moment of Y can be obtained by the similar way. Since the algebra is rather tedious, it is omitted here for convenience. Alternatively,
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Unformatted text preview: the &rst and second moments of Y can be obtained by the Taylor±s expansion of exponential functions, i.e. e bt = 1 + bt + ( bt ) 2 2 + ( bt ) 3 6 + ²²² and e at = 1 + at + ( at ) 2 2 + ( at ) 3 6 + ²²² : Then, the MGF of Y becomes M Y ( t ) = 1 b ± a ³ ( b ± a ) + b 2 ± a 2 2 t + b 3 ± a 3 6 t 2 + ²²² ´ : 1 The &rst moment of Y is given by E ( Y ) = dM Y ( t ) dt & & & & t =0 = b 2 & a 2 2 ( b & a ) = b + a 2 : The second moment of Y is given by E ( Y 2 ) = d 2 M Y ( t ) dt 2 & & & & t =0 = 2( b 3 & a 3 ) 6( b & a ) = a 2 + ab + b 2 3 : Hence, the variance of Y is given by V ar ( Y ) = a 2 + ab + b 2 3 & ± b + a 2 ² 2 = ( b & a ) 2 12 : 2...
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Chapter 4 Continuous Random Variables Appendix (First and Second Moments of Uniform Random Variable)

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