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1) Let the random variables ? be the sum of independent Poisson distributed random variables, i.e.,...

1) Let the random variables ? be the sum of independent Poisson distributed random variables, i.e., ? = ∑ ? (top) ?=1(bottom) ?? , where ?? is Poisson distributed with mean ?? .

(a) Find the moment generating function of ?? . (b) Derive the moment generating function of ?. (c) Hence, find the probability mass function of ?.

2)The moment generating function of the random variable X is given by ??(?) = exp{7(?^(?)) − 7} and that of ? by ?? (?) = ( (8/9) (?^(?)) + (6/9))^10 . Assuming that ? and ? are independent, find

(a) ?{? + ? = 7}.

(b) ?{?? = 0}.

(c) ?(??).

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