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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85% of tax returns have computational errors. This means that if we randomly select a tax return, there is a probability of 0.85 that the tax return will have at least one computational error. Whether one tax return has a computational error is independent of whether any other tax return has a computational error. John Jay, a new tax return auditor randomly selects tax returns for audit one after another. Let X = the number of tax returns selected until he selects the 1st return with computational errors. Let Y = the number of tax returns selected until he selects the 2nd return with computational errors. What is the probability that the 1st 3 tax returns selected have computational errors? What is the expected value of X? What is the variance of X? What is the probability that X > 3? What is the probability that X < 3? What is the probability that Y = 4? What is the probability that Y< 4?
A club recruit member either through active recruitment by its current members or through advertising in neighborhood billboards. Suppose that each member of the club recruit new members at a rate of 3 members per week and that advertising in neighborhood billboards generate new members at a rate of 2 members per week. Suppose further that each member is expected to stay a club member for 1 week, on average. Assume that the number of members of the club at any given time follows a birth and death process. Given that the club started with 5 members, what is the expected number of club members after the first two weeks? (Hint: refer to the theorem below) Theorem For a linear growth process with immigration and Xo = i, we have 0 · (e(x-μ)t - 1) + ie(x-μ)t, x‡μ E (X₂) = x-fl Ot + i, λ = μ.
PART C. 1. James is an operator of a machine that uses 16 batteries. Because it has so many, it does not shut off until half of the batteries are dead. a. All batteries are independent and have uniformly distributed lifetimes on the interval [ 0, 1 2] years. Find the probability that the machine shuts before 1 years of work? b. What is the expected time when the 5th and the 8th battery die?
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 login password consists of 4 letters followed by 2 numbers. Assume that the password is Case-sensitive. What is the probability that randomly chosen password begins with the letters AB?
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?
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?
Assume that the batter hits with probability 0.2 against left-handed pitchers, and with probability 0.3 against right-handed pitchers. Both situations are Bernoulli processes. One month the player bats 22 times against left-handed pitchers and 52 times against right-handed pitchers. How many hits should the batter expect to get during this month?
3. A coin is tossed repeatedly, heads occurring on each toss with probability p. Find the probability generating function of the number T of tosses before a run of n heads has appeared for the first time.
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?
A drawer holds purple socks and yellow socks. If n socks are taken out of the drawer at random, the probability that all are yellow is 1/2. What is the smallest possible number of socks in the drawer (as a function of n)?
Suppose that {u¡ t = ...,-2,-1,0,1,2,...} is an independent time series with mean 0 variance o = 4.0. Suppose that the time series {xj: t = ...,-2,-1,0,1,2,...} satisfies the equation.: x=0.5 x_₁ + 1.0+ u, -0.4 -1. Determine the 1. mean, 2. autocovariance function, 3. variance, 4. autocorrelation function, 5. Partial autocorrelation function of the time series x₁.

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