Let X1, ... , Xn be iid with the Poisson(λ) distribution. What is the conditional distribution of Xi given the sample mean?
Let X1, ... , Xn be iid with the Poisson(λ) distribution. What is the conditional distribution...
Let X1, …, Xn be iid Poisson(λ). Find the maximum likelihood estimator λMLE for λ, when it is given that λ > λ*, where λ* > 0 is a fixed constant. (Note: This is asking you to find the restricted MLE)
4. Let Xi, , Xn be iid sample from a Poisson population with parameter λ. (a) Construct an confidence interval for λ by inverting an LRT (b) The following data, the number of aphids per row in nine rows of a potato field, can be assumed to follow a Poisson distribution: 155, 104, 66, 50, 36, 40, 30, 35, 42. Use these data to construct a 90% LRT confidence interval for the mean number of aphids per row.
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,...
Let X1, . . . , Xn ∼ iid Exp(λ) and Y1, . . . , Ym ∼ iid Exp(τ ) be independent random samples. (a) Find the restricted MLEs under the null hypothesis H0 : λ = τ . (b) Write out a formula for the LRT statistic, and describe how you could perform this test asymptotically.
(a)Suppose X ∼ Poisson(λ) and Y ∼ Poisson(γ) are independent, prove that X + Y ∼ Poisson(λ + γ). (b)Let X1, . . . , Xn be an iid random sample from Poisson(λ), provide a sufficient statistic for λ and justify your answer. (c)Under the setting of part (b), show λb = 1 n Pn i=1 Xi is consistent estimator of λ. (d)Use the Central Limit Theorem to find an asymptotic normal distribution for λb defined in part (c), justify...
Suppose X1,. , Xn are iid Poisson(A) random variables. Show by direct calculation without using any theoremm in mathematical statistics, that (a) Ση! Xi/n is an unbiased estimator for λ. (b) X is optimal in MSE among all unbiased estimators. This is to say, let T be another unbiased estimator, then EA(X) EA(T2
Suppose X1 and X2 are iid Poisson(θ) random variables and let T = X1 + 2X2. (a) Find the conditional distribution of (X1,X2) given T = 7. (b) For θ = 1 and θ = 2, respectively, calculate all probabilities in the above conditional distribution and present the two conditional distributions numerically.
Again, let X1,..., Xn be iid observations from the Uniform(0,0) distribution. (a) Find the joint pdf of Xi) and X(n)- (b) Define R = X(n)-X(1) as the sample range. Find the pdf of R. (c) It turns out, if Xi, . . . , xn (iid) Uniform(0,0), E(R)-θ . What happens to E® as n increases? Briefly explain in words why this makes sense intuitively.
6. Let X1, . . . , Xn denote a random sample (iid.) of size n from some distribution with unknown μ and σ2-25. Also let X-(1/ . (a) If the sample size n 64, compute the approximate probability that the sample mean X n) Σηι Xi denote the sample mean will be within 0.5 units of the unknown p. (b) If the sample size n must be chosen such that the probability is at least 0.95 that the sample...
Let X1, ..., Xn denote an independent random sample from a population with a Poisson distribution with mean . Derive the most powerful test for testing Ho : 1= 2 versus Ha: 1= 1/2.