Homework 5

# Homework 5 - University of California Los Angeles...

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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 . In other words, as λ → ∞ , the ratio Z = X - λ λ converges to the standard normal distribution.

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