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Let X,, z . 2. Estimate Xn be ind poisson random variobles with parometors P(120 5...
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) =...
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) = P(X1 = -1) = 1/2. Let Sn = X1 + X2 + ... + xn. (a) Find ES, and var(Sn); (b) Show that Sn is a martingale.
Let Xi, ..., Xn be random variables with the same mean and with covariance function where...
Let Xi, ..., Xn be random variables with the same mean and with covariance function where |ρ| < 1 . Find the mean and variance of Sn-Xi + . . . + Xn. Assume thatE(X. ) μ and V(X) σ2 for i (1.2. , n}
Problem 3 Let Xi, X2,... , Xn be a sequence of binary, i.i.d. random variables. Assume...
Problem 3 Let Xi, X2,... , Xn be a sequence of binary, i.i.d. random variables. Assume P (Xi 1) P (Xi = 0) = 1/2. Let Z be a parity check on seluence Xi, X2, ,X,, that is, Z = X BX2 e (a) Is Z statistically independent of Xi? (Assume n> 1) (b) Are X, X2, ..., Xn 1, Z statistically independent? (c) Are X, X2,.., Xn, Z statistically independent? (d) Is Z statistically independent of Xi if P...
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let...
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let Zn-(Sn-np)/ V np (1-p). Show that as n oo, Zn converges to the standard normal RV Z~ N(0,1) in distribution. (ii) Conclude that if Yn~Binomial(n, p), then (iii) From i, deduce that have the following approximation x-np which becomes more accurate as n → oo.
5. Let Xi, . . . , Xn be a random sample from f(x:0) = -| for z > 0. (a) Assume that θ 0.2 Using ...
5. Let Xi, . . . , Xn be a random sample from f(x:0) = -| for z > 0. (a) Assume that θ 0.2 Using the Inversion Method of Sampling, write a R function to generate data from f(x; 0). (b) Use your function in (a) to draw a sample of size 100 from f(0 0.2 (c) Find the method of moments estimate of θ using the data in (b). (d) Find the maximum likelihood estimate of θ using...
Please let me know how to solve 7.6.5. 6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with p...
Please let me know how to solve 7.6.5.
6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with parameter θ > 0. (a) Find the MVUE of P(X < 1)-(1 +0)c". Hint: Let u(x)-1, where Y = Σ1Xi. 1, zero elsewhere, and find Elu(Xi)|Y = y, xỉ (b) Express the MVUE as a function of the mle of θ. (c) Determine the asymptotic distribution of the mle of θ (d) Obtain the mle of P(X...
2. Let X1, ..., Xn be a random sample from a Poisson random variable X with...
2. Let X1, ..., Xn be a random sample from a Poisson random variable X with parameter 1 (a) Derive the method of moments estimator of 1 (4 marks) (b) Derive the maximum likelihood estimator of 1 (4 marks) (c) Derive the least squares estimator of 1 (4 marks) (d) Propose an estimator for VAR[X] (4 marks) (e) Propose an estimator for E[X²] (4 marks)
3. Let Xi, . . . , Xn be random samples of X and X(1), ....
3. Let Xi, . . . , Xn be random samples of X and X(1), . . . , X(n) ordered random samples of X which are obtained from a rearrangement of X1,... , Xn such that (a) Show that the empirical distribution functions of Xi,..., Xn and Xo),..., X(n) coincide. (b) Consider the samples taken from X ~ F. Use (a) to compute A-2), F,,(-1)Ћ,(1.8),Ћ,(25) (c) (Continued from (b)) Plot A,(z) over-2 4.
Problem 8 (4x4 pts) Suppose Xi, X2-, ..,. Xn are each independent Poisson random variables with...
Problem 8 (4x4 pts) Suppose Xi, X2-, ..,. Xn are each independent Poisson random variables with mean 1. Let 100 k=1 (a) Rccall, Markov's incquality is P(Y > a) for a> 0 Using Markov's inequality, estimate the probability that P(Y > 120). (b) Rccal, Chebyshev's incquality is Using Chebyshev's inequality, estimate P( Y-?> 20) (c), (d) Using the Central Limit Theorem, estimate P(Y > 120) and Ply-? > 20).
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each oth...
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) = P(X1 = -1) = 1/2. Let Sn = X1 + X2 + ... + xn. (a) Find ES, and var(Sn); (b) Show that Sn is a martingale.
Let Xi, ..., Xn be random variables with the same mean and with covariance function where |ρ| < 1 . Find the mean and variance of Sn-Xi + . . . + Xn. Assume thatE(X. ) μ and V(X) σ2 for i (1.2. , n}
Problem 3 Let Xi, X2,... , Xn be a sequence of binary, i.i.d. random variables. Assume P (Xi 1) P (Xi = 0) = 1/2. Let Z be a parity check on seluence Xi, X2, ,X,, that is, Z = X BX2 e (a) Is Z statistically independent of Xi? (Assume n> 1) (b) Are X, X2, ..., Xn 1, Z statistically independent? (c) Are X, X2,.., Xn, Z statistically independent? (d) Is Z statistically independent of Xi if P...
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let Zn-(Sn-np)/ V np (1-p). Show that as n oo, Zn converges to the standard normal RV Z~ N(0,1) in distribution. (ii) Conclude that if Yn~Binomial(n, p), then (iii) From i, deduce that have the following approximation x-np which becomes more accurate as n → oo.
5. Let Xi, . . . , Xn be a random sample from f(x:0) = -| for z > 0. (a) Assume that θ 0.2 Using the Inversion Method of Sampling, write a R function to generate data from f(x; 0). (b) Use your function in (a) to draw a sample of size 100 from f(0 0.2 (c) Find the method of moments estimate of θ using the data in (b). (d) Find the maximum likelihood estimate of θ using...
Please let me know how to solve 7.6.5.
6.5. Let Xi, X2,. .. X, be a random sample from a Poisson distribution with parameter θ > 0. (a) Find the MVUE of P(X < 1)-(1 +0)c". Hint: Let u(x)-1, where Y = Σ1Xi. 1, zero elsewhere, and find Elu(Xi)|Y = y, xỉ (b) Express the MVUE as a function of the mle of θ. (c) Determine the asymptotic distribution of the mle of θ (d) Obtain the mle of P(X...
2. Let X1, ..., Xn be a random sample from a Poisson random variable X with parameter 1 (a) Derive the method of moments estimator of 1 (4 marks) (b) Derive the maximum likelihood estimator of 1 (4 marks) (c) Derive the least squares estimator of 1 (4 marks) (d) Propose an estimator for VAR[X] (4 marks) (e) Propose an estimator for E[X²] (4 marks)
3. Let Xi, . . . , Xn be random samples of X and X(1), . . . , X(n) ordered random samples of X which are obtained from a rearrangement of X1,... , Xn such that (a) Show that the empirical distribution functions of Xi,..., Xn and Xo),..., X(n) coincide. (b) Consider the samples taken from X ~ F. Use (a) to compute A-2), F,,(-1)Ћ,(1.8),Ћ,(25) (c) (Continued from (b)) Plot A,(z) over-2 4.
Problem 8 (4x4 pts) Suppose Xi, X2-, ..,. Xn are each independent Poisson random variables with mean 1. Let 100 k=1 (a) Rccall, Markov's incquality is P(Y > a) for a> 0 Using Markov's inequality, estimate the probability that P(Y > 120). (b) Rccal, Chebyshev's incquality is Using Chebyshev's inequality, estimate P( Y-?> 20) (c), (d) Using the Central Limit Theorem, estimate P(Y > 120) and Ply-? > 20).
8.12. In the zero-inflated Poisson model, random data xi...xn are assumed to be of the form xrii where the y have a Poi(a) distribution and the have a Ber(p) distribution, all independent of each other. Given an outcome x-(xi, , X.), the objective is to estimate both λ and p. Consider the following hierarchical Bayesian model: P U(0, 1) alp) Gammala, b) rlp.i)~Ber(p independently (x,lr.λ.Ρ) ~ Poiar.) independently . where r () and a and b are known parameters. We...
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