1. Consider a branchi ng process with offspring distribution (p) k2o- Let X be the number...
9. Consider the Branching Process {Xn,n = 0,1,2,3,...} where Xn is the population size at the nth generation. Assume P(Xo = 1) = 1 and that the probability generating function of the offspring distribution is common A(z) (z3322z + 4) 10 (а) If gn 3 P(X, — 0) for n %3D 0, 1,..., write down an equation relating ^n+1 and qn. 0,1,2 Hence otherwise, evaluate qn for n= or (b) Find the extinction probability q = lim00 n 6 marks]...
(1) Consider the following processes: There are No = 1 many individuals in the zeroth generation. The number of individuals N in the kth generation comes from each individual in the (k-1)th generation having Poisson(A) many offspring independent of all others. (a) Find a formula for E(Nk). (b) Suppose X1. Show that P(Nk 0) converges to unity as ko N. = 0) converg (2) Consider the processes from the previous problem modified so that the number of offspring which each...
Let X0, X1, X2,... be a
branching process (as defined in class), i.e. Xn gives
then number of individuals in the nth generation. Suppose that the
mean number of offspring per individual is μ. Show that
Mn = μ-nXn is a martingale with
respect to X0, X1, X2,...
Let Xo, X1, X2,... be a branching process (as defined in class), i.e., Xn gives then number of individuals in the nth generation. Suppose that the mean number of offspring per individual...
lain what is meant by the term 'branching process . (Uxford 1974) e nth generation of a branching process in which eac 6. (b) Let Xn be the size of th has probability generating function G, and assume generating function Gn of Xn satisfies Gni(s) - 1. Show fXn satisfies Gn+1 (s) = Gn(G(s)) forn>that the probabili m variable when ar, B (O, 1), and find GIn explicitly wh ß is the probability generating function of a non-ne (c) Show,...
1. Let X have probability generating function Gx (s) and let un generating function U(s) of the sequence uo, u1, ... satisfies P(X > n). Show that the (1- s)U(s) = 1 - Gx(s), whenever the series defining these generating functions converge.
1. Let X have probability generating function Gx (s) and let un generating function U(s) of the sequence uo, u1, ... satisfies P(X > n). Show that the (1- s)U(s) = 1 - Gx(s), whenever the series defining...
Let X be a random variable which follows truncated binomial distribution with the following p.m.f. P(X=x) =((n|x)(p^x)(1−p)^(n−x))/(1−(1−p)^n), if x= 1,2,3,···,n. •Find the moment generating function (m.g.f.) and the probability generating function(p.g.f.). •From the m.g.f./p.g.f., and/ or otherwise, obtain the mean and variance. Show all the necessary steps for full credit.
7. Let X a be random variable with probability density function given by -1 < x < 1 fx(x) otherwise (a) Find the mean u and variance o2 of X (b) Derive the moment generating function of X and state the values for which it is defined (c) For the value(s) at which the moment generating function found in part (b) is (are) not defined, what should the moment generating function be defined as? Justify your answer (d) Let X1,...
2. Let(Zn : n = 0,1,2, be a branching process with offspring distribution X ~ Bin(2, ). That is, A r X=x | 0 | 1 | 2 | Total P(X = r) | 16 | 16 | 16 Find the probabhility that the process will eventually extinct
(4) Consider the following random family tree: Let Y, denote the (random) number of people in the nth generation. Each person in the nth generation produces a random number of offspring, which has a Poisson(A) distribution. The total number of such children is then denoted Yn1 The number of offspring produced by any person is (statistically) independent of the number produced by another person. Moreover, Yo 1, that is, there is exactly one person in the zeroth generation. (a) Determine...
Consider the following probability distribution. X 0 2 4 6 P(X = x) 1/4 1/4 1/4 1/4 3. (5 points) Suppose we draw n random samples (X1, ... , Xn), and an estimator 0(X1, ... , Xn) is proposed as n B(x,,,X,) X;I(Xi 70, and X; #6), n i=1 where I(-) is an indicator function, I(X; # 0, and Xi # 6) = 0, if X; € {0,6}, and I(X; # 0, and X; + 6) = 1, if Xi...