Estimate posterior mean θ with Poisson likelihood for the exponential prior with the prior mean E(θ) = μ = 2 and the data vector x = (3,1,4,3,2).
Estimate posterior mean θ with Poisson likelihood for the exponential prior with the prior mean E(θ)...
Question 2 a. Show that, for the exponential model with gamma prior, the posterior Π(9121m) under n observations can be computed as the posterior given a single observation xn using the prior q(の는 1101r1:n-1). Give the formula for the parameters (an,ßn) of the posterior ll(θ|X1:n, α0,Ao) as a function of (an-1, Bn-1). b. Visualize the gradual change of shape of the posterior II(01:n, ao, Bo) with increasing n: . Generate n 256 exponentially distributed samples with parameter θ-1. . Use...
what is the likelihood function of Poisson density that is proportional to the posterior density?
QUESTION B 5. (a) Describe the Dirichlet-Multinomial model. In particular, define the Likelihood, the prior and posterior distributions. Additionally, suggest a non-informative prior. 5 marks) ) A sample of n 1000 women were interviewed about their preferences on three brand of eyeliners. The outcome is the following: 1st Brand234 preferences 338 preferences 2nd Brand 428 preferences 3rd Brand Suppose that this dataset is analyzed with a Dirichlet-Multinomial model. Estimate the posterior probabilities of selecting a certain brand when a Dir(40,20,30)...
Random variable X corresponds to the daily number of accidents in a small town during the first week of January. From the previous experience (prior infor- mation), local police Chief Smith tends to believe that the mean daily number of accidents is 2 and the variance is also 2. We also observe for the current year the sample number of accidents for 5 days in a row: 5,2,1,3,3. Let us assume that X has Poisson distribution with parameter θ ....
3. This problem is concerned with the maximum likelihood estimate (MLE) of various distributions. Bob, Céline and Daisy want to model the distribution of the heights of 20 students in the classroom. They get the following data (in cm) : 168, 177, 194, 169, 159, 172, 174, 177, 159, 172, 181, 171, 168, 162, 168, 157, 180, 174, 162, 177. (i) Bob took Math170A, and he wants to model the heights by the normal distribution with probability density p(x) e...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
just part a plz thank u! Page 4 Marks Suppose Z(t) Y., where X(t) is the Poisson process with rate θ If μ = E[h] and σ2-Yar determine the mean and variance of Z(t) a. pil are the common mean and variance for y,y , then b. fVis Uniform distribution on interval (0, 1], then determine the mean and variance of XCV) 2 Page 4 Marks Suppose Z(t) Y., where X(t) is the Poisson process with rate θ If μ...
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...
Let Y1, Y2, ..., Yn denote a random sample from an exponential distribution with mean θ. Find the rejection region for the likelihood ratio test of H0 : θ = 2 versus Ha : θ ≠ 2 with α = 0.09 and n = 14. Rejection region =
Let Xi , i = 1, · · · , n be a random sample from Poisson(θ) with pdf f(x|θ) = e −θ θ x x! , x = 0, 1, 2, · · · . (a) Find the posterior distribution for θ when the prior is an exponential distribution with mean 1; (b) Find the Bayesian estimator under the square loss function. (c) Find a 95% credible interval for the parameter θ for the sample x1 = 2, x2...