2birthdeath

# 2birthdeath - Birth Processes Birth-Death Processes...

This preview shows pages 1–9. Sign up to view the full content.

Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Birth-death processes Jorge J´ulvez University of Zaragoza 1 / 47

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Outline 1 Birth Processes 2 Birth-Death Processes 3 Relationship to Markov Chains 4 Linear Birth-Death Processes 5 Examples 2 / 47
Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Outline 1 Birth Processes 2 Birth-Death Processes 3 Relationship to Markov Chains 4 Linear Birth-Death Processes 5 Examples 3 / 47

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Example: Consider cells which reproduce according to the following rules: A cell present at time t has probability λ h + o ( h ) of splitting in two in the interval ( t , t + h ) This probability is independent of age Events betweeen different cells are independent Time > 4 / 47
Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Example: Consider cells which reproduce according to the following rules: A cell present at time t has probability λ h + o ( h ) of splitting in two in the interval ( t , t + h ) This probability is independent of age Events betweeen different cells are independent Time > What is the time evolution of the system? 4 / 47

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Non-Probabilistic Analysis Let n ( t ) = number of cells at time t Let λ be the birth rate per single cell Thus λ n ( t )Δ( t ) births occur in ( t , t + Δ t ) Then: n ( t + Δ t ) = n ( t ) + n ( t ) λ Δ t n ( t + Δ t ) - n ( t ) Δ t = n ( t ) λ dn dt = n 0 ( t ) = n ( t ) λ The solution of this differential equation is: n ( t ) = Ke λ t If n ( 0 ) = n 0 then n ( t ) = n 0 e λ t 5 / 47
Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Probabilistic Analysis Notation: N ( t ) = number of cells at time t P { N ( t ) = n } = P n ( t ) Assumptions: A cell present at time t has probability λ h + o ( h ) of splitting in two in the interval ( t , t + h ) The probability of more than one birth occurring in time interval ( t , t + h ) is o ( h ) All states are transient 6 / 47

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Birth Processes Birth-Death Processes Relationship to Markov Chains Linear Birth-Death Processes Examples Pure Birth Process (Yule-Furry Process) Assumptions: Probability of splitting in ( t , t + h ) : λ h + o ( h ) Probability of more than one split in ( t , t + h ) : o ( h ) The probability of birth in ( t , t + h ) if N ( t ) = n is n λ h + o ( h ) .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern