Examples of Univariate Transformations.pdf

Hint do not compute the sum if then 1 1 1 this is the

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trials and probability of success given by 1− p. [Hint: do NOT compute the sum. If 𝑀𝑀 𝑌𝑌 ( 𝑡𝑡 ) = 𝐸𝐸 [ 𝐸𝐸 𝑌𝑌𝑌𝑌 ], then 𝐸𝐸 [ 𝑒𝑒 −𝑌𝑌𝑌𝑌 ] = 𝑀𝑀 𝑌𝑌 (? )] 𝑀𝑀 𝑈𝑈 ( 𝑡𝑡 ) = 𝐸𝐸 [ 𝑒𝑒 𝑈𝑈𝑌𝑌 ] = 𝐸𝐸�𝑒𝑒 ( 𝑛𝑛−𝑌𝑌 ) 𝑌𝑌 = 𝐸𝐸 [ 𝑒𝑒 𝑛𝑛𝑌𝑌 𝑒𝑒 −𝑌𝑌𝑌𝑌 ] = 𝑒𝑒 𝑛𝑛𝑌𝑌 𝐸𝐸�𝑒𝑒 𝑌𝑌 ( −𝑌𝑌 ) = 𝑒𝑒 𝑛𝑛𝑌𝑌 𝑀𝑀 𝑌𝑌 ( −𝑡𝑡 ) = 𝑒𝑒 𝑛𝑛𝑌𝑌 (1 − 𝑝𝑝 ) + 𝑝𝑝 𝑒𝑒 −𝑌𝑌 𝑛𝑛 = (1 − 𝑝𝑝 ) 𝑒𝑒 𝑌𝑌 + 𝑝𝑝� 𝑛𝑛 = ( 𝑝𝑝 + (1 − 𝑝𝑝 ) 𝑒𝑒 𝑌𝑌 ) 𝑛𝑛 This is the moment generating function of 𝐵𝐵𝐵𝐵𝐵𝐵𝐹𝐹𝑚𝑚𝐵𝐵𝐵𝐵𝑒𝑒 ( 𝐵𝐵 , 1 − 𝑝𝑝 ) distribution.
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  • Winter '08
  • moulib
  • Probability theory, Cumulative distribution function

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