a) Compute E(Zn+1| Zn = k) for k ≥ 0. b) Compute E(Zn+1|Zn). c) Compute E(Zn). d) What can you say about P(Zn 0) as n→ ∞ when <1? [Use Markov's inequality.] What does this mean for the population? e) What do you expect might happen when µ > 1? =
a) Compute E(Zn+1| Zn = k) for k ≥ 0. b) Compute E(Zn+1|Zn). c) Compute E(Zn). d) What can you say about P(Zn 0) as n→ ∞ when <1? [Use Markov's inequality.] What does this mean for the population? e) What do you expect might happen when µ > 1? =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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![Problem 8.
Consider a randomly growing population in which there are
Zn individuals in the population at generation n for n = 0, 1, 2,.... Each individual in generation
n gives birth to a random number of offspring: say the ith individual in the nth generation gives
birth to X) individuals in generation n + 1, where {X(n) : i ≥ 1, n ≥ 0} are independent and
identically distributed random variables, each with range {0, 1, 2,...}. Then we can define Zn
recursively:
Zn+1 =
[x(n) + ... + X(n)
0
and we suppose the population starts with Zo
the mean number of offspring per individual in the population is E(X(")) = µ.
a) Compute E(Zn+1|Zn = k) for k ≥ 0.
b) Compute E(Zn+1|Zn).
c) Compute E(Zn).
d) What can you say about P(Zn
What does this mean for the population?
e) What do you expect might happen when μ> 1?
=
if Zn = k for k ≥ 1
if Zn = 0,
=
1, a single individual at generation 0. Suppose
0) as n when μ< 1? [Use Markov's inequality.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71ca2b49-cfa8-4243-95a5-ec7901b35b1a%2Fd5ce6730-bc7b-462b-a0d3-3b6651df589f%2Fxlukyfvr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 8.
Consider a randomly growing population in which there are
Zn individuals in the population at generation n for n = 0, 1, 2,.... Each individual in generation
n gives birth to a random number of offspring: say the ith individual in the nth generation gives
birth to X) individuals in generation n + 1, where {X(n) : i ≥ 1, n ≥ 0} are independent and
identically distributed random variables, each with range {0, 1, 2,...}. Then we can define Zn
recursively:
Zn+1 =
[x(n) + ... + X(n)
0
and we suppose the population starts with Zo
the mean number of offspring per individual in the population is E(X(")) = µ.
a) Compute E(Zn+1|Zn = k) for k ≥ 0.
b) Compute E(Zn+1|Zn).
c) Compute E(Zn).
d) What can you say about P(Zn
What does this mean for the population?
e) What do you expect might happen when μ> 1?
=
if Zn = k for k ≥ 1
if Zn = 0,
=
1, a single individual at generation 0. Suppose
0) as n when μ< 1? [Use Markov's inequality.]
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