An immigration-birth process with arrival rateλand birth rateβmay be described by the probability statement

P(X(t +δt)= x +1| X(t)= x)=(λ+βx)δt+0(δt)

Suppose that at time 0 the size of a population growing according to the above rule is 4. An ‘event’ is said to have occurred when the population size increases by one (and it is immaterial whether this increase is due to a random arrival or to a birth).

Write down an expression for the expected waiting time until thenth event (that is until the population size reachesn+ 4).

Since X(0) = 4,

should the birthrate beβor 4β? I am confused here.

I am leaning towards 4β, so am I correct that the expected waiting time should be:

1/(λ+ 4β) + 1/(λ+ 8β) + 1/(λ+12β) + ... + 1/(λ+ 4nβ)

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If the above is correct, for the p.g.f. of a simple birth process with X(0) = 2 and birthrate β, should it be:

Π(s, t) = [(se^(-βt))/(1-s(1-e^(-βt)))]^2

or

Π(s, t) = [(se^(-2βt))/(1-s(1-e^(-2βt)))]^2

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Using the former (which I think is correct), is the following workings to find the probabilities at X(t) = 3 correct?

Π(3, t) = [(3e^(-βt))/(1-3(1-e^(-βt)))]^2

= (9e^(-2βt))/(-2+3e^(-βt))^2

= (9e^(-2βt))/(4-12e^(-βt)+9e^(-2βt))