hexaedru8p

2022-10-02

Suppose that lions have a geometric offspring distribution. ${p}_{k}=(1-\theta ){\theta}^{k}$, $k\ge 0$ with mean 1.5. If the current world population of lions is 10, what is the probability they will go extinct in 1 generation? What is the probability they will go extinct eventually?

Same two questions whose offspring distribution is Poisson with mean 0.8, and current population is 5000.

Same two questions whose offspring distribution is Poisson with mean 0.8, and current population is 5000.

Caiden Brewer

Beginner2022-10-03Added 5 answers

Step 1

Suppose every member gives birth independently to an identically distributed number of "children", X is the number of offspring. If $P(X=0)>0$

$P(the\text{}process\text{}will\text{}go\text{}extinct)=1\phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}\text{}\mathbb{E}X\le 1$

Step 2

When $\mathbb{E}X>1$, and you want to compute P(the process will go extinct), then you need to use the generating function for X.

Suppose every member gives birth independently to an identically distributed number of "children", X is the number of offspring. If $P(X=0)>0$

$P(the\text{}process\text{}will\text{}go\text{}extinct)=1\phantom{\rule{thickmathspace}{0ex}}\u27fa\phantom{\rule{thickmathspace}{0ex}}\text{}\mathbb{E}X\le 1$

Step 2

When $\mathbb{E}X>1$, and you want to compute P(the process will go extinct), then you need to use the generating function for X.

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