Compute extinction probability in a branching process
Problem Description
Consider a population of cells that begins with 1 cell at time 0. Every minute, each existing cell independently undergoes one of four possible transitions, each with an equal probability of 1/4:
- Dies: The cell produces 0 new cells.
- Stays the same: The cell produces 1 new cell.
- Divides into 2 cells: The cell produces 2 new cells.
- Divides into 3 cells: The cell produces 3 new cells.
Let (Z_t) denote the total number of cells in the population after (t) minutes.
Questions
- What is the probability that the population is extinct at (t=2) minutes? That is, calculate (P(Z_2=0)).
- What is the probability that the population eventually goes extinct at some finite time (t)? That is, calculate (P(\exists t: Z_t=0)).
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Accurately modeling and predicting the long-term behavior of a stochastic process that can either explode or die out, requiring both precise analytical methods and an understanding of computational feasibility and scalability for real-world applications.
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Compute extinction probability in a branching process
Problem Description
Consider a population of cells that begins with 1 cell at time 0. Every minute, each existing cell independently undergoes one of four possible transitions, each with an equal probability of 1/4:
- Dies: The cell produces 0 new cells.
- Stays the same: The cell produces 1 new cell.
- Divides into 2 cells: The cell produces 2 new cells.
- Divides into 3 cells: The cell produces 3 new cells.
Let (Z_t) denote the total number of cells in the population after (t) minutes.
Questions
- What is the probability that the population is extinct at (t=2) minutes? That is, calculate (P(Z_2=0)).
- What is the probability that the population eventually goes extinct at some finite time (t)? That is, calculate (P(\exists t: Z_t=0)).