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(Solved): (a) A discrete random variable x has the following probability mass function: p_(x)(t)={(C_(1)2^(-t) ...
(a) A discrete random variable x has the following probability mass function:
p_(x)(t)={(C_(1)2^(-t), for t=1,3,5,7,dots),(0, otherwise ):}
where C_(1) is a constant.
(i) Find the value of C_(1).
(ii) Derive the moment generating function of x. State the domain of the function.
(iii) Find E(x) and Var(x).
(b) A discrete random variable Y has the following probability mass function:
p_(Y)(t)={(C_(2)2^(-t), for t=2,4,6,8,dots;),(0, otherwise ):}
where C_(2) is a constant.
(i) Find the value of C_(2).
(ii) Derive the moment generating function of Y. State the domain of the function.
(iii) Find E(Y) and Var(Y).