Branching with immigration. Each generation of a branching process (with a single progenitor)is augmented by a random number of immigrants who are indistinguishable from the other membersof the population. Suppose that the numbers of immigrants in different generations are independentof each other and of the past history of the branching process, each such number having probabilitygenerating function H(s). Show that the probability generating function Gn of the size of the nthgeneration satisfies Gn+1(s) = Gn(G(s))H(s), where G is the probability generating function of atypical family of offspring. Branching with immigration. Each generation of a branching process (with a single progenitor)is augmented by a random number of immigrants who are indistinguishable from the other membersof the population. Suppose that the numbers of immigrants in different generations are independentof each other and of the past history of the branching process, each such number having probabilitygenerating function H(s). Show that the probability generating function Gn of the size of the nthgeneration satisfies Gn+1(s) = Gn(G(s))H(s), where G is the probability generating function of atypical family of offspring.

Note.... Same question is giving two times . Giving Answer will be the same as they are not…

Answered: Branching with immigration. Each…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Suppose that 55% of all babies born in a particular hospital are girls. If 7 babies born in the hospital are randomly selected, what is the probability that at least 2 of them are girls? Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places. (If necessary, consult a list of formulas.) X
2
This continuous-time process with state space N starts in state 0 and repeats in the following way. It remains in the same state for some random waiting time at which point the state number increases by some random increment. The waiting times are independent and all follow the same fixed probability distribution taking values in Ro. The increments are independent (and also independent of the waiting times) and all follow another fixed probability distribution taking values in {1, 2, 3, ... ..}. So, if the waiting times are S₁, S2, ... and the increments are n₁, n₂,... then we start with X (0) = 0 and remain with X(t) = 0 for t = [0, S₁). Then we jump up by n₁ so X(t) = n₁ for t € [S₁, S₁ + S₂). Then, at time S₁ + S2, we jump up to n₁ + n₂ and so on. More formally, for t € [S₁ + … + Sk, S₁ + + Sk+ Sk+1) we have X(t) = n₁ + + nk. (a) Which choices of distribution for waiting time and increment show that the Poisson Process is a special case of this kind of process?
A system consists of five components is connected in series as shown below. 5 3 As soon as one component fails, the entire system will fail. Assume that the components fail independently of one another. (a) Suppose that each of the first two components have lifetimes that are exponentially distributed with mean 110 weeks, and that each of the last three components have lifetimes that are exponentially distributed with mean 132 weeks. Find the probability that the system lasts at least 49 weeks. (b) Now suppose that each component has a lifetime that is exponentially distributed with the same mean. What must that mean be (in years) so that 97% of all such systems lasts at least one year?
Pacific salmon populations have discrete breeding cycles in which they return from the ocean to streams to reproduce and then die. This occurs every one to five years, depending on the species. (a) Suppose that each fish must first survive predation by bears while swimming upstream, and predation occurs with probability d. After swimming upstream, each fish produces b offspring before dying. The stream is then stocked with m additional newly hatched fish before all fish then swim out to sea. What is the discrete-time recursion for the population dynamics, assuming that there is no mortality while at sea? You should count the population immediately before the upstream journey. nt + 1 = (b) Suppose that, instead of preying on fish while they swim upstream, bears do so only while the fish are swimming downstream. What is the discrete-time recursion for the population dynamics? (Again assume there is no mortality while at sea.) nt+1 (c) Which of the recursions obtained in parts (a) and (b)…
QUESTION (3): A Factory produces packets of papers. The number of papers per packet varies as shown in the following table. NUMBER OF PAPERS 97 98 99 100 101 102 103 PROPORTION OF PACKETS .04 .13 .21 .29 .20 .10 .03 Draw the probability mass function (PMF). а. b. Calculate and draw the cumulative probability funetion (PDF). What is the probability that a randomly chosen packet will contain between 99 and 101 papers? с. d. Two packets are chosen at random. What is the probability that at least one of them contains at least 100 papers?
A man and a woman agree to meet at a certain location about 12:30 P.M. If the man arrives at a time uniformly distributed between 12:15 and 12:45, and if the woman independently arrives at a time uniformly distributed between 12:00 and 1 P.M., find the probability that the first to arrive waits no longer than 5 minutes. What is the probability that the man arrives first?
Assume that Tom attends class randomly with probability 0.55 and that each decision is independent of previous attendance, so that the process can be viewed as a Bernoulli process. What is the probability that he attends at least 7 of 10 classes given that he attends at least 2 but not all 10 classes?
In a certain insurance has 20 holders. The ages of these policy holders are as follows: One is 16 years old, four are 18, nine are 19, three are 20, two are 21, and one is 30. Let X= age of any policy holder (randomly selected). Find the probability mass function of age.
Pacific salmon populations have discrete breeding cycles in which they return from the ocean to streams to reproduce and then die. This occurs every one to five years, depending on the species. (a) Suppose that each fish must first survive predation by bears while swimming upstream, and predation occurs with probability d. After swimming upstream, each fish produces b offspring before dying. The stream is then stocked with m additional newly hatched fish before all fish then swim out to sea. What is the discrete-time recursion for the population dynamics, assuming that there is no mortality while at sea? You should count the population immediately before the upstream journey. nt + 1 = (b) Suppose that, instead of preying on fish while they swim upstream, bears do so only while the fish are swimming downstream. What is the discrete-time recursion for the population dynamics? (Again assume there is no mortality while at sea.) nt+1 = (c) Which of the recursions obtained in parts (a) and…
5c The genotype of an organism can be either normal (wild type, W) or mutant (M). Each generation, a wild type individual has probability 0.03 of having a mutant offspring, and a mutant has probability 0.005 of having a wild type offspring. Calculate the probability of a string of the following genotypes in successive generations: WWW.

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