Data yi, i = 1, . . . , n arise from a Poisson distribution with...
Data yi, i = 1, . . . , n arise from a Poisson distribution with rate parameter λ.
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(a) Show that the posterior distribution for λ|y1,...,yn is Gamma distributed when a Gamma(α,β) prior is used.
(b) If the data are: y = 17, 25 , 25 , 21 , 13 , 22 , 23; find the posterior for λ given the above specified Gamma prior.
Homework Answers
The Poisson distribution is a useful discrete distribution which can be used to model the number ...
PROBABILITY QUESTION
The Poisson distribution is a useful discrete distribution which can be used to model the number of occur rences of something per unit time. If X is Poisson distributed, i.e. X Poisson(λ), its probability mass function takes the following form: oisson distributed, i.e. X - Assume now we have n identically and independently drawn data points from Poisson(A) :D- {r1,...,Xn Question 3.1 [5 pts] Derive an expression for maximum likelihood estimate (MLE) of λ. Question 3.2 5pts Assume...
Please answer the question clearly. Consider a random sample of size n from a Poisson population...
Please answer the question clearly.
Consider a random sample of size n from a Poisson population with parameter λ (a) Find the method of moments estimator for λ. (b) Find the maximum likelihood estimator for λ. Suppose X has a Poisson distribution and the prior distribution for its parameter A is a gamma distribution with parameters and β. (a) Show that the posterior distribution of A given X-x is a gamma distribution with parameters a +r and (b) Find the...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
For all of following, calculate the A) Posterior Distribution B)Bayes estimator of θ C) Predictive probability...
For all of following, calculate the A) Posterior Distribution B)Bayes estimator of θ C) Predictive probability 1) yi iid∼ Bern(θ), i = 1, . . . , n1, and yj iid∼ Bern(2θ), j = n1 + 1, . . . , n1 + n2, yi and yj mutually independent . Use θ ∼ Beta(α, β) for prior 2) Same as problem 1 but with Bin(Mi,θ) and Bin(Mi, 2θ) instead of Bern(θ) and Bern(2θ), respectively. Use θ ∼ Beta(α, β) for...
Suppose the number of events that happen in time t follows a Poisson distribution with parameter...
Suppose the number of events that happen in time t follows a Poisson distribution with parameter λ. Show that the time when the third even occurs follows a gamma distribution with α = 3,β = 1/λ.
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...
Let Y1,K,Y n denote a random sample from a Poisson distribution with parameter λ . a....
Let Y1,K,Y n denote a random sample from a Poisson distribution with parameter λ . a. Find a sufficient statistics for λ. b. Find the minimum variance unbiased estimator(MVUE) of λ2 .
Problem 1- Bias and variance for the Poisson distribution This is a thought ezperiment of a kind statisticians often do. Imagine that... n data points ri:m are sampled 1.1.d. from a Poisson distribut...
Problem 1- Bias and variance for the Poisson distribution This is a thought ezperiment of a kind statisticians often do. Imagine that... n data points ri:m are sampled 1.1.d. from a Poisson distribution with parameter λ (and because it's a thought erperiment, we assume we know λ.) Recall also that λΜ L-Ση1r/n. a. Calculate E[AML] as a function of λ. All erpectations are under the true distribution of the data. b. Calculate Var(AML) as a function of λ.
Problem 1-...
Consider the simplified Bayesian model for normal data The joint posterior pdf is ful, σ21 x)a(σ2,-/2-1 expl_jy.tx, _aP...
Consider the simplified Bayesian model for normal data The joint posterior pdf is ful, σ21 x)a(σ2,-/2-1 expl_jy.tx, _aPI The marginal posterior pdfs of μ and σ 2 can be obtained by integrating out the other variable (8.30) y@1 x) α (σ2)-m;,-1/2 expl-- Σ.-tri-x)2 (8.31) d. Let q1 and q2 be they/2 and 1-y/2 quantiles of (8.31). Show that the 1-γ credible interval (gi,q2) is identical to the classic confidence interval (5.19) (with ar replaced by y). Hence, a (1-α) stochastic...
Let X1, . . . , Xn be a sample taken from the Gamma distribution Γ(2,...
Let X1, . . . , Xn be a sample taken from the Gamma distribution Γ(2, θ−1) with pdf f(x,θ)= θ^2xexp(−θx) if x ≥ 0, θ ∈ (0,∞), and 0 otherwise, (A) Show that Y = ∑ni=1 Xi is a complete and sufficient statistic. (B) Find E(1/Y) . Hint: If W ∼ χ2(k) then E(W^m) = 2mΓ(k/2+m) for m > −k/2. Note also that Y Γ(k/2) Γ(n) = (n − 1)!, n ∈ N∗ . Facts from 1(C) are useful:...
PROBABILITY QUESTION
The Poisson distribution is a useful discrete distribution which can be used to model the number of occur rences of something per unit time. If X is Poisson distributed, i.e. X Poisson(λ), its probability mass function takes the following form: oisson distributed, i.e. X - Assume now we have n identically and independently drawn data points from Poisson(A) :D- {r1,...,Xn Question 3.1 [5 pts] Derive an expression for maximum likelihood estimate (MLE) of λ. Question 3.2 5pts Assume...
Please answer the question clearly.
Consider a random sample of size n from a Poisson population with parameter λ (a) Find the method of moments estimator for λ. (b) Find the maximum likelihood estimator for λ. Suppose X has a Poisson distribution and the prior distribution for its parameter A is a gamma distribution with parameters and β. (a) Show that the posterior distribution of A given X-x is a gamma distribution with parameters a +r and (b) Find the...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
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...
Problem 1- Bias and variance for the Poisson distribution This is a thought ezperiment of a kind statisticians often do. Imagine that... n data points ri:m are sampled 1.1.d. from a Poisson distribution with parameter λ (and because it's a thought erperiment, we assume we know λ.) Recall also that λΜ L-Ση1r/n. a. Calculate E[AML] as a function of λ. All erpectations are under the true distribution of the data. b. Calculate Var(AML) as a function of λ.
Problem 1-...
Consider the simplified Bayesian model for normal data The joint posterior pdf is ful, σ21 x)a(σ2,-/2-1 expl_jy.tx, _aPI The marginal posterior pdfs of μ and σ 2 can be obtained by integrating out the other variable (8.30) y@1 x) α (σ2)-m;,-1/2 expl-- Σ.-tri-x)2 (8.31) d. Let q1 and q2 be they/2 and 1-y/2 quantiles of (8.31). Show that the 1-γ credible interval (gi,q2) is identical to the classic confidence interval (5.19) (with ar replaced by y). Hence, a (1-α) stochastic...
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