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1. An immigration-birth process with...

1. An immigration-birth process with arrival rate λ and birth rate β may 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).

What is the expected waiting time until the first event?

2. An immigration-birth process with arrival rate λ and birth rate β may 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 the nth event (that is until the population size reaches n + 4).

3. An immigration-birth process with arrival rate λ and birth rate β may 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 variance of this waiting time until the nth event (that is until the population size reaches n + 4).

4. Suppose a simple birth process with birth rate β starts with two individuals. If Wn is the time to the nth birth, the c.d.f of W2 is 1 - 3e^(-2βt) + 2e^(-3βt).

Is it true or false?

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).

What is the expected waiting time until the first event?

2. An immigration-birth process with arrival rate λ and birth rate β may 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 the nth event (that is until the population size reaches n + 4).

3. An immigration-birth process with arrival rate λ and birth rate β may 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 variance of this waiting time until the nth event (that is until the population size reaches n + 4).

4. Suppose a simple birth process with birth rate β starts with two individuals. If Wn is the time to the nth birth, the c.d.f of W2 is 1 - 3e^(-2βt) + 2e^(-3βt).

Is it true or false?

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