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Question 5 – 8 marks This...

Question 5 – 8 marks

This question is intended to assess your understanding of birth and death

processes.

You should be able to answer this question after working through Part I of

Book 4.

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}.

A partial diﬀerential equation for Π(s,t), the probability generating

function of X(t), is

∂Π ∂t = −βs(1 − s) ∂ ∂s .

page 17 of 31In Activity 4.1 of Book 4, you showed that this partial diﬀerential equation

has the general solution

Π(s,t) = ψ $ s 1 − s e −βt)

.

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 {2,3,...}

and parameters r = 2 and p = 0.2. Find the particular solution

corresponding to this initial condition. Hence identify the probability

distribution of X(t) in this case, and ﬁnd its mean.

http://www.mediafire.com/view/?y14kmm74a4q964n

This question is intended to assess your understanding of birth and death

processes.

You should be able to answer this question after working through Part I of

Book 4.

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}.

A partial diﬀerential equation for Π(s,t), the probability generating

function of X(t), is

∂Π ∂t = −βs(1 − s) ∂ ∂s .

page 17 of 31In Activity 4.1 of Book 4, you showed that this partial diﬀerential equation

has the general solution

Π(s,t) = ψ $ s 1 − s e −βt)

.

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 {2,3,...}

and parameters r = 2 and p = 0.2. Find the particular solution

corresponding to this initial condition. Hence identify the probability

distribution of X(t) in this case, and ﬁnd its mean.

http://www.mediafire.com/view/?y14kmm74a4q964n

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