Suppose that X is a random variable for which the moment generating function is as follows: MX(t) = e^(2t^2+3t) for −∞ < t < ∞. Find the mean and variance of X. (b) Suppose that X has moment generating function MX(t) =(3e^t/4 + 1/4)^6 (i) Find the p.m.f. of X.

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Chapter1: Combinatorial Analysis
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Suppose that X is a random variable for which the moment generating function is as follows:
MX(t) = e^(2t^2+3t) for −∞ < t < ∞.
Find the mean and variance of X.
(b) Suppose that X has moment generating function MX(t) =(3e^t/4 + 1/4)^6
(i) Find the p.m.f. of X.
(ii) Find the mean and variance of X.
(c) A person with some finite number of keys wants to open a door. He tries the keys one-by-one independently at random with replacement. How many trails you expect, from him, to open the door?
(d) Obtain the form of moment generating function (m.g.f.) for the following p.m.f. –
p(x) = ((2^x)(e^-2))/×!, x = 0,1,2, … .

Also calculate the mean and variance from m.g.f.

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