Markov Chain — Branching Process

– Stochastic Course Notes

1. Definition

Consider a population

  • Each individual produces j new offspring each period with probability p_j, j geq 0. Key assumption: This distribution is constant over time.
  • Assume that p_j < q for all j (i.e., the problem is not deterministic)
  • Let X_n be the size of the population at period n

2. Number of Population in the n-th Generation

3. Probability of Die Out

Analysis

Example

4. Example: Family Genetics

Problem:

Solution: 


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