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The integer-valued random variable X(t) denotes the number of individuals alive at time t in a simple birth process {X (t);t 2 0). A partial differential equation for II(s, t), the probability generating function of X (t), is 하 =-88(1-8) on Os Ot In Activity 4.1 of Book 4, you showed that this partial differential equation has the general solution Suppose that X(0), the number of individuals alive at time 0, is a random variable: X(0) has the negative binomial distribution with range (4,5,...) and parameters r 4 and p 0.8. Find the particular solution corresponding to this initial condition. Hence identify the probability distribution of X(t) in this case, and find its mean
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