ESM3A HW10 - Jacobs University Bremen School of Engineering...

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Unformatted text preview: Jacobs University Bremen School of Engineering and Science Peter Oswald Fall Term 2010 120201 ESM3A — Problem Set 10 Issued: 18.11.2010 Due: Wednesday 24.11.2010 (in class) This problem set deals with sequences of random variables, and limit theorems. 10.1. Another important function associated with a random variable X is the moment-generating function M X ( t ) defined by M X ( t ) := E ( e tX ) = ∞ X n =0 E ( X n ) n ! t n , where the finiteness of the expectation E ( e tX ) resp. the convergence of the power series is assumed to hold in some open interval t ∈ (- δ,δ ) with δ > 0. Under modest assumptions, knowing M X ( t ) allows to identify the distribution and density functions of X in a unique way. For the exponential distribution with parameter λ , find M X ( t ), investigate the associated power series, and determine the moments E ( X n ) for all n ≥ 1. 10.2. Consider a sequence of independent random variables X i , i = 1 , 2 ,... ....
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