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5. In a discrete time population branching process, the probability that an individual has j offspring...
9. Consider the Branching Process {Xn,n = 0,1,2,3,...} where Xn is the population size at the nth generation. Assume P(...
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]...
lain what is meant by the term 'branching process . (Uxford 1974) e nth generation of...
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,...
Question related to branching processes. Zn is the number of offspring in generation n. I know...
Question related to branching processes. Zn is the number of
offspring in generation n. I know that Pk is a geometric
distribution, but am unsure of where to go from there.
Exercise 9.17 Find the mean and variance of Zn when the family-size distribution is given by P for k 0, 1, 2, . . . , and 0 < p 뉘-q < 1 . Deduce that var(Zn)-0 if and only if p
Given the discrete uniform population with parameter N = 8, find the probability that a random...
Given the discrete uniform population with parameter N = 8, find the probability that a random sample of size 36, selected with replacement, will produce a sample mean greater than 5. Answer using 4 decimals.
Given the discrete uniform population with parameter N = 8, find the probability that a random...
Given the discrete uniform population with parameter N = 8, find the probability that a random sample of size 36, selected with replacement, will produce a sample mean greater than 5. Write the answer with 4 decimals.
Suppose that a given individual in a population has a survival time which is exponential with...
Suppose that a given individual in a population has a survival time which is exponential with a hazard rate 0. Each individual's hazard rate θ s potentially different and is sampled from a gamma distribution with density function TCB) Let X be the life length of a randomly chosen member of this popula- tion. (a) Find the survival function of X. (Hint: Find S(x) Ele" .) (b) Find the hazard rate of X. What is the shape of the hazard...
2. One common distribution that appears in branching process theory is a DRV with pmf: fx(x;j)...
2. One common distribution that appears in branching process theory is a DRV with pmf: fx(x;j) = w. e-h (ux)2–1 — where x E {1, 2, ...} and pi € [0, 1] a. Find the MLE for u given iid X1, ..., Xn. Then, find the MLE for the particular data X1 = 2, X2 = 1, X3 = 6. b. Using Desmos (https://www.desmos.com/calculator), draw a graph of the likelihood function (not log-likelihood) for the data x1 = 2, X2...
The waiting time T between successive occurrences of an event E in a discrete-time renewal process has the probability distribution P(T- 2)0.5 and P(T 3)-0.5. a) Find the generating function U(s) for...
The waiting time T between successive occurrences of an event E in a discrete-time renewal process has the probability distribution P(T- 2)0.5 and P(T 3)-0.5. a) Find the generating function U(s) for this process and hence or otherwise find the [4 probabilities u, us and e (b) The waiting time to the fifth renewal is denoted by W (i) Find the range of Ws (ii) Find the probability P(Ws- 13).
The waiting time T between successive occurrences of an event...
5. A stationary random process X[n] is input to a discrete time LTI system with frequency...
5. A stationary random process X[n] is input to a discrete time LTI system with frequency response j“)-10 zero mean given as A(e nmay be expressed as where Wnlis a zero mea a-HS1 unit variancei.i.d. (independent identically distributed) Gaussian sequence and c, d are constants. Let Yl be the output random a)Determine the mean function for the output random sequence Yn in terms ofa, c and d b) Determine S7 (e), the power spectral density ofthe output random sequence Yn]...
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]...
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,...
Question related to branching processes. Zn is the number of
offspring in generation n. I know that Pk is a geometric
distribution, but am unsure of where to go from there.
Exercise 9.17 Find the mean and variance of Zn when the family-size distribution is given by P for k 0, 1, 2, . . . , and 0 < p 뉘-q < 1 . Deduce that var(Zn)-0 if and only if p
Suppose that a given individual in a population has a survival time which is exponential with a hazard rate 0. Each individual's hazard rate θ s potentially different and is sampled from a gamma distribution with density function TCB) Let X be the life length of a randomly chosen member of this popula- tion. (a) Find the survival function of X. (Hint: Find S(x) Ele" .) (b) Find the hazard rate of X. What is the shape of the hazard...
2. One common distribution that appears in branching process theory is a DRV with pmf: fx(x;j) = w. e-h (ux)2–1 — where x E {1, 2, ...} and pi € [0, 1] a. Find the MLE for u given iid X1, ..., Xn. Then, find the MLE for the particular data X1 = 2, X2 = 1, X3 = 6. b. Using Desmos (https://www.desmos.com/calculator), draw a graph of the likelihood function (not log-likelihood) for the data x1 = 2, X2...
The waiting time T between successive occurrences of an event E in a discrete-time renewal process has the probability distribution P(T- 2)0.5 and P(T 3)-0.5. a) Find the generating function U(s) for this process and hence or otherwise find the [4 probabilities u, us and e (b) The waiting time to the fifth renewal is denoted by W (i) Find the range of Ws (ii) Find the probability P(Ws- 13).
The waiting time T between successive occurrences of an event...
5. A stationary random process X[n] is input to a discrete time LTI system with frequency response j“)-10 zero mean given as A(e nmay be expressed as where Wnlis a zero mea a-HS1 unit variancei.i.d. (independent identically distributed) Gaussian sequence and c, d are constants. Let Yl be the output random a)Determine the mean function for the output random sequence Yn in terms ofa, c and d b) Determine S7 (e), the power spectral density ofthe output random sequence Yn]...
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