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M 1: Suppose that.X, form a random sample from a Bernoulli distribution for s unknown (0 θ < 1). ...
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to...
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to be taken from a Bermoulli distribution for which the value of the parameter θ is unknown, and the prior distribution of θ is a Beta(α,β) distribution. Represent the mean of this prior distribution as μο=α/(α+p). The posterior distribution of θ is Beta =e+ ΣΧ, β.-β+n-ΣΧ.) a) Show that the mean of the posterior distribution is a weighted average of the form where yn and...
Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ...
Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ 0. În turn, the parameter 0 is best regarded as a random variable with a Pareto(a, b) distribution, that is, bab 0, otherwise, where a 〉 0 and b 〉 0 are known. (a) Turn the "Bayesian crank" to find the posterior distribution of θ. I would probably start by working with a sufficient statistic (b) Find the posterior mean and use this as...
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+ ), where θ e (-00, Exercise...
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+ ), where θ e (-00, Exercise 7.5: Suppose X1, X2, . .. , sufficient for θ. a) Show that the smallest and largest of Xi, ..., Xn are jointliy (b) If p@-constant, θ e (-00, oo), is the prior distribution of θ, find its posterior distribution
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+...
Question 3: Bernoulli distribution (23/100 points) Consider a random sample X1,...,Xn from a Bernoulli distribution with...
Question 3: Bernoulli distribution (23/100 points) Consider a random sample X1,...,Xn from a Bernoulli distribution with unknown parameter p that describes the probability that Xi is equal to 1. That is, Bernoulli(p), i = 1, ..., n. (10) The maximum likelihood (ML) estimator for p is given by ÔML = x (11) n It holds that NPML BIN(n,p). (12) 3.a) (1 point) Give the conservative 100(1 – a)% two-sided equal-tailed confidence interval for p based on ÔML for a given...
explan the answer 10: A certain continuous distribution has cumulative distribution function (CDF) given by F(r)...
explan the answer
10: A certain continuous distribution has cumulative distribution function (CDF) given by F(r) 0, <0 where θ is an unknown parameter, θ > 0. (i) Find (a) the p.d.f., (b) the mean and (e) the variance of this distribution. (ii) Suppose that X (Xi, X2, Xn) is a random sample from this distribu- tion and let Y max(Xi, XXn). Find the CDF and p.d.f. of Y. Hence find the value of a for which EloY)
Number 4 turns out to be an inverse gamma function with parameters alpha= n and beta=...
Number 4 turns out to be an inverse gamma function with
parameters alpha= n and beta= the sum of x sub i
PLEASE ANSWER #5 NOT #4
4. Suppose that X1,X2, 10 pts. the p.d.f. is given by form a random sample from a distribution for which where the unknown parameter θ > 0. Suppose also that the improper prior of θ is m(0) Find the posterior distribution π(θ x). Hint: The inverse gamina distribution from question 6 in Homework...
3. Prove the theorem for t he normal conjugate distributi on Theorem. Suppose that Xi,... ,Xn...
3. Prove the theorem for t he normal conjugate distributi on Theorem. Suppose that Xi,... ,Xn form a random sample from a normal distribution for which the value of the mean θ is unknown and the value of the variance σ2 > 0 is known. Suppose also that the prior distribution of θ is the normal distribution with mean 140 and variance v . Then the posterior distribution of θ given that Xi-Xi,1-1, . . . , n, is the...
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ ...
difficult……
2and4 thanks
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ ...
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ. (b) Find an...
3. Let X and 6 be two random variables. Let X given 0 have a Bernoulli...
3. Let X and 6 be two random variables. Let X given 0 have a Bernoulli distribution with parameter θ, this is, X | θ ~ Bernoulli (9), and let θ have a beta distribution with parameters a and b, this is 9 Beta(a, b), where a and b are known positive constants (a) Find the joint distribution of (X,6 (b) Find the marginal distribution of X.
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to be taken from a Bermoulli distribution for which the value of the parameter θ is unknown, and the prior distribution of θ is a Beta(α,β) distribution. Represent the mean of this prior distribution as μο=α/(α+p). The posterior distribution of θ is Beta =e+ ΣΧ, β.-β+n-ΣΧ.) a) Show that the mean of the posterior distribution is a weighted average of the form where yn and...
Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ 0. În turn, the parameter 0 is best regarded as a random variable with a Pareto(a, b) distribution, that is, bab 0, otherwise, where a 〉 0 and b 〉 0 are known. (a) Turn the "Bayesian crank" to find the posterior distribution of θ. I would probably start by working with a sufficient statistic (b) Find the posterior mean and use this as...
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+ ), where θ e (-00, Exercise 7.5: Suppose X1, X2, . .. , sufficient for θ. a) Show that the smallest and largest of Xi, ..., Xn are jointliy (b) If p@-constant, θ e (-00, oo), is the prior distribution of θ, find its posterior distribution
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+...
Question 3: Bernoulli distribution (23/100 points) Consider a random sample X1,...,Xn from a Bernoulli distribution with unknown parameter p that describes the probability that Xi is equal to 1. That is, Bernoulli(p), i = 1, ..., n. (10) The maximum likelihood (ML) estimator for p is given by ÔML = x (11) n It holds that NPML BIN(n,p). (12) 3.a) (1 point) Give the conservative 100(1 – a)% two-sided equal-tailed confidence interval for p based on ÔML for a given...
explan the answer
10: A certain continuous distribution has cumulative distribution function (CDF) given by F(r) 0, <0 where θ is an unknown parameter, θ > 0. (i) Find (a) the p.d.f., (b) the mean and (e) the variance of this distribution. (ii) Suppose that X (Xi, X2, Xn) is a random sample from this distribu- tion and let Y max(Xi, XXn). Find the CDF and p.d.f. of Y. Hence find the value of a for which EloY)
Number 4 turns out to be an inverse gamma function with
parameters alpha= n and beta= the sum of x sub i
PLEASE ANSWER #5 NOT #4
4. Suppose that X1,X2, 10 pts. the p.d.f. is given by form a random sample from a distribution for which where the unknown parameter θ > 0. Suppose also that the improper prior of θ is m(0) Find the posterior distribution π(θ x). Hint: The inverse gamina distribution from question 6 in Homework...
3. Prove the theorem for t he normal conjugate distributi on Theorem. Suppose that Xi,... ,Xn form a random sample from a normal distribution for which the value of the mean θ is unknown and the value of the variance σ2 > 0 is known. Suppose also that the prior distribution of θ is the normal distribution with mean 140 and variance v . Then the posterior distribution of θ given that Xi-Xi,1-1, . . . , n, is the...
difficult……
2and4 thanks
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ....
Mathematical Statistics แ (Homework y 5) 1. Let , be a random sample fiom the densit where 0 s θ 1 . Find an unbiased estimator of Q 2. Let Xi, , x. be independent random variables having pdfAx; t) given by Show that X is a sufficient statistic for e f(xl A) =-e- . x > 0 3. Let Xi, , x,' be a random sample from exponential distribution with (a) Find sufficient statistic for λ. (b) Find an...
3. Let X and 6 be two random variables. Let X given 0 have a Bernoulli distribution with parameter θ, this is, X | θ ~ Bernoulli (9), and let θ have a beta distribution with parameters a and b, this is 9 Beta(a, b), where a and b are known positive constants (a) Find the joint distribution of (X,6 (b) Find the marginal distribution of X.
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