# branching

1. ### Branching process Galton-watson basic question

Let {Xn : n ≥ 0} be a Galton-Watson branching process starting with one particle (i.e., X0 = 1) and offspring distribution p0 = 1,/6 p1 = 1/3, p2 = 1/3, p3 = 1/6. Find E(Xn), Var(Xn) and P (Population ever dies out). Thank you :) I'm lost :(
2. ### Branching process with martingale

Let \{ X_k^n : n,k \geq 1 \} be i.i.d. positive interger-value random variables with EX_k^n = \mu < \infty and Var(X_k^n) = \sigma ^2 > 0 . Define Y_0 =1 and recursively define Y_{n+1}=X^{n+1}_1+ . . . +X_{Y_n}^{n+1} \ \ \ n \geq 0 a) Show that M_n = \frac {Y_n}{ \mu ^n } is a...
3. ### Variance of a branching process

Let N_j be the size of the jth generation of a branching process with N_0 = 1. Suppose the number of offspring produced by an individual has mean \mu and variance {\sigma}^2. Let \alpha_j = \frac{Var(N_j)}{{{\sigma}^2}{\mu}^{j-1}} Show that \alpha_j = \alpha_{j-1}+\mu^{j-1} I get to...
4. ### Galton-Walton branching process

In a Galton-Walton branching process, the offspring distribution is given by \mathbb{P}(X = r) = p_0 \ if \ r=0, (1-p_0)pq^{r-1}\if\ r=1,2,... where 0 < p < 1, q=1-p, and \ 0 < p_0 < 1 Find the probability of extinction (assuming N_0=1) i did: find the generating function first...
5. ### Age dependent branching process

Let Z_n be the size of the nth generation in an age-dependent branching process Z(t), the lifetime distribution of which is exponential with parameter \lambda. If Z(0) = 1, show that the probability generating function G_t (s) of Z(t) satisfies \frac{\partial}{\partial t} G_t(s) = \lambda...
6. ### Markov Chain: Branching Process

Branching Process.
7. ### Branching Process

Ive done part a and b,ive done most of c as all you need to do is write down the laws of generating functions but im not sure what to do for d and e
8. ### Branching Process

Hi all, i am trying to understand branching processes but struggling.... on this question for instance.... Consider a branching process in which the distribution of the number of offspring of a single individual is given by: P(Za = 0) = 0.25, P(Za = 1) = 0.5 and P(Za = 2) = 0.25 What is the...
9. ### Another branching process problem

Check this link: Probability and random processes - Google Books The problem I am trying to complete is #6, page 175 from this book. Any ideas?
10. ### Age-dependent branching process

Consider a branching process with generation sizes Zn satisfying Z0 = 1 and P(Z1 = 0) = 0. Pick two individuals at random (with replacement) from the nth generation and let L be the index of the generation which contains their most recent common ancestor. Show that: P(L = r) = E[...
11. ### Branching Process

Consider a branching process {Xn:n=0,1,...} with X0~Poisson(λ) and an offspring distribution which is binomial(4,p) where p>1/4. Calculate the probability of ultimate extinction.
12. ### Branching Process and Generating Functions

How do you find Gn(s) given G1(s)? I know Gn(s) = G(G(...G(s))) n times but I'm unable to work out the algebra. Please use G1(s) = (2-s)^-1 as an example. Thanks!
13. ### PGF of a branching process (very hard)

Let X_0, X_1, ... be a branching process with \mathbb{P} (X_0 = 1) = 1, and offspring distribution \mathbb{P} (Y= k ) = qp^k \ \mbox{for k} = 0,1,2,... where q = 1- p \ \mbox{and} \ p \in (0,1). (1) Prove that the pgf G_n \ \mbox{of} \ X_n is G_n (s) = \begin{cases}...
14. ### Branching Process.

Think I might need to refresh myself in power series before doing this one. Anyway, how should I proceed with this question? Consider the Branching Process {X_n, n = 0, 1, 2, 3, ...} where X_n is the population size of the nth generation. Assume P(X_o = 1) = 1 and that the pgf of the common...
15. ### Branching Processes

I have managed to complete part (a) of this question but i dont have an idea of how to do part (b). i have included my attempt of the beginning of part (b) but i do not know where to go from there or if i am even on the right track. If anyone has any idea please help! Thanks
16. ### Branching Process

P0=.5, P1=.1,P3=.4 Suppose X0=1 then What is the prob. that the population is extinct in the third generation, given that it was not extinct in the second generation? It looks simple, but I am getting different answer from my classmates... Thank you~~!
17. J

### Analysis: Branching process

I missed a few classes and I have an assignment due soon. I missed the classes that covered this area. Can anyone help point me in the right direction or show me how to do this?