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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Homework 5 EXERCISES 1 Find the distribution of the random variable X for each of the following moment-generating func- tions: a. M X ( t ) = h 1 3 e t + 2 3 i 5 . b. M X ( t ) = e t 2- e t . c. M X ( t ) = e 2( e t- 1) . EXERCISES 2 Let M X ( t ) = 1 6 e t + 2 6 e 2 t + 3 6 e 3 t be the moment-generating function of a random variable X . a. Find E ( X ). b. Find var ( X ). c. Find the distribution of X . EXERCISES 3 Let X follow the Poisson probability distribution with parameter λ . Its moment-generating function is M X ( t ) = e λ ( e t- 1) . a. Show that the moment-generating function of Z = X- λ √ λ is given by: M Z ( t ) = e- √ λt e λ ( e t √ λ- 1) . b. Use the series expansion of e t √ λ = 1 + t √ λ 1! + ( t √ λ ) 2 2! + ( t √ λ ) 3 3! + ··· to show that lim λ →∞ M Z ( t ) = e 1 2 t 2 ....
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This note was uploaded on 01/14/2011 for the course STATS 100A taught by Professor Wu during the Fall '07 term at UCLA.
- Fall '07