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Probability : Random Variable with Binomial Distribution, Random Walk Process and Probability Generating Function

A1. Suppose that a random variable X has a Binomial distribution with parameters n and p. Find E[(1/3)^X].

A2. In a particular type of random walk process, a particle starts from 0 at time 0, and then, at each time 1,2,3,..., either moves one step to the right with probability p or stays at the same position with probability q = 1? p. If we denote by Sn the distance to the right which the particle has moved from 0 after n steps, then S0 = 0 and, assóming that the moves are independent of each other,

Sn = Σ j =1..n Xj n= 1,2,3....

where X1, X2,. .. are independent, identically distributed random variables such that
P(Xj=1)=p and P(Xj=0)=q= 1- p.
(i) Given n = 1, 2,..., what is the distribution of Sn?
(ii) For n = 1, 2, . .., what is the expected position of the particle at time n?

A3. Consider a random variable X with probability function Px = P(X = x), x = 0, 1,2,....
and probability generating function Gx (s).
(i) Define the random variable Y = aX + b, where a and b are positive integers. If Gy(s) is the probability generating function of Y, show that
Gy(s) = sbGx (sa)
(ii) If px = 1/2^x+1, x = 0, 1, 2,.. ., determine Gx(s) and Gy(s). Use Gy(s) to show that the mean of Y is a+b.

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Random Variable with Binomial Distribution, Random Walk Process and Probability Generating Function are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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