session7 - [STAT 100A/SANCHEZ TA SESSION 7 Practice moment...

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[STAT 100A/SANCHEZ TA SESSION] 7 1 Practice moment generating functions. 1. For the following densities, find the constant c, the moment generating function M(t), the cumulative distribution function F, and the third moment (Hint: you may need to use L’Hospital rule at some point). (a) f(X) = c a x b (b) f(X)= ce -x x>0 The moment-generating function is not differentiable at zero, but the moments can be calculated by differentiating and then taking limit t 0. lim t 0 ʹ′ ʹ′ ʹ′ M ( t ) = 1 4 ( a + b )( a 2 + b 2 ) ( a )1 = c dx a b = cx a b | = c ( b a ) c = 1 b a F ( x ) = P ( X x ) = 1 b a dx = x a b a a x F ( x ) = 1 x > b F ( x ) = 0 x < a M ( t ) = E [ e tX ] = e tx 1 b a Λ Ν Μ Ξ Π Ο dx a b = 1 b a e tx dx a b = 1 b a 1 t e tx Ρ Σ ΢ Τ Φ Υ a b = 1 ( b a ) t ( e tb e ta ) ʹ′ M ( t ) = e bt ( bt 1) e at ( at 1) ( b a ) t 2 ʹ′ ʹ′ M ( t ) = ( b a ) t 3 e bt b 2 e at a 2 [ ] 2( b a ) t e bt ( bt 1) e at ( at 1) [ ] ( b a ) t 4 = t 2 e bt b 2 e at a 2 [ ] 2 e bt ( bt 1) e at ( at 1) [ ] t 3 = e bt b 2 t 2 2 bt + 2 ( ) e at ( a 2 t 2 2 at + 2) t 3 ʹ′ ʹ′ ʹ′ M ( t ) = t 3 e bt b 3 e at a 3 [ ] 3 e bt b 2 t 2 2 bt + 2 ( ) e at ( a 2 t 2 2 at + 2) [ ] t 4
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2 ( b )1 = ce x dx 0 = c e x dx 0 = c e x [ ] 0 = c c = 1 F ( x ) = P ( X x ) = e t dt = 1 e x 0 x M ( t ) = e tx e x dx = e ( t
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