Question

Consider a population in which there are no deaths and each individual independently gives birth at rate 1. Independently of the other events, immigrants arrive in the population at times of a Poisson process with rate 2. Let X(t) denote the number of individuals in the population at time t. The pure birth process (X(t),t ≥ 0) is called a Yule process with immigration.

(a) Give the birth rates for the process (X(t),t ≥ 0).

(b) Find P(X(2) = 1|X(0) = 1) and P(X(2) = 3|X(1) = 3).

(c) Assuming that there is one individual at time zero, find the expected value and variance of the amount of time that it takes before the population reaches size 4.

Answer 1

Answer:

given by:

a)The birth rates for the process (x(t),t>0) by:

The Markov property comes as of the memorylessness of the exponential distribution intended for event times.

This is a linear birth/death process with immigration, have parameters µn = nµ and

λn = nλ + θ.

1n = 1n+2, n = n + 2

(b) P(X(2) = 1|X(0) = 1) and P(X(2) = 3|X(1) = 3)by:

P(X(2) = 1|X(0) = 1) = P(X(0) = 1, X(2) = 1)/P(X(0) = 1)

=

= 1/2= 0.5

P(X(2) = 3|X(1) = 3) = P(X(1) = 3, X(2) = 3)/P(X(1) = 3)

= =2/3=0.666

(c) The expected value and variance of the amount of here by:

assume that there is one individual at time 0,

E[X(t + h)|X(0)] = E[X(t)|X(0)] + (λ − µ) E[X(t)|X(0)]h + θh + o(h)

Then,

Defining by

‘’M(t) ≡ E[X(t)|X(0)]’’

we find the differential equation by

‘’M'(t) = (λ − µ)M(t) + θ’’

  the initial condition with M(0) = i,

The expected value and variance of the amount of time that it takes before the population reach size 4

M(t) = θt + i if λ = µ (i + θ λ−µ )e (λ−µ)t − θ λ−µ otherwise