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: