Suppose integer valued random variable with pgf

module web page following the submission deadline. You are expected to work independently and note that disciplinary action will be taken for plagiarism.

1. Suppose W and X are independent random variables with respective PGFs: GW (θ) = eθ−1,
GX(θ) = 0.1 + 0.4θ + 0.5θ2.

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3. At The University of Exeter the frequency of cars passing the Peter Chalk Centre is recorded. It is noticed that Mercedes, BMWs and Ferraris all go passed as Poisson Processes with re-spective rates of 5, 2, and 1 per hour. You may assume that these Poisson Processes are all independent.

4. Suppose X is an integer valued random variable with PGF:
GX(θ) = α + αθ + (1 − 2α)θk,
where α ∈ (0, 1/2) is a constant, and k ≥ 3. Suppose we have a branching process, where Sn denotes the number of individuals at stage n (notation as in lectures), and S0 = 1. You may assume that the number of offspring X arising from any individual in any generation has a probability distribution governed by GX(θ), and that individuals evolve independently of each other.

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