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# Random Variables, Probability Mass Functions, Probability Generating Functions and Offspring Generating Functions

A1. (i) Let X be a random variable with probability mass function ( pmf )
.....
Determine the probability generating function of X.
(ii) Suppose that the probability generating function of a random variable X is given by
....
Determine the probability mass function of X. [5 marks]
(Hint: You may use the formula: P1
...
A2. Let Zn, n &#8805; 0 be a branching process with offspring distribution
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(i) Find the offspring generating function G(s) and the expectation of N. [4 marks]
(ii) Determine the probability generating function of Z2 and write down P(Z2 = 3).

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keywords: pmf, p.m.f., pgf, p.g.f.

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Problem A1
(1) Since the probability mass function of is , , then the probability generating function of is
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#### Solution Summary

Random Variables, Probability Mass Functions, Probability Generating Functions and Offspring Generating Functions are investigated. The solution is detailed and well presented.

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