(a) The integer-valued random variable X(t) denotes the number of individuals alive at time t in...

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(a) The integer-valued random variable X(t) denotes the number of individuals alive at time t in a simple birth process {X(t); t ≥ 0} with birth rate ß. The partial differential equation for its probability generating function II(s, t) is ап =-Bs(1-8)- Ət which has the general solution given by II(s, t) = * (₁²-²). (i) Given that this population starts with five individuals, find a particular solution for the population. ап Əs (ii) Identify the probability distribution of X(t) in this case, and find its mean. (iii) Let W₂ be the waiting time to the second birth. Obtain the cumulative distribution function of W₂. [2] (a) The integer-valued random variable X(t) denotes the number of individuals alive at time t in a simple birth process {X(t); t ≥ 0} with birth rate ß. The partial differential equation for its probability generating function II(s, t) is ап =-Bs(1-8)- Ət which has the general solution given by II(s, t) = * (₁²-²). (i) Given that this population starts with five individuals, find a particular solution for the population. ап Əs (ii) Identify the probability distribution of X(t) in this case, and find its mean. (iii) Let W₂ be the waiting time to the second birth. Obtain the cumulative distribution function of W₂. [2]