4. Let Xi, , Xn be iid sample from a Poisson population with parameter λ. (a)...
Let X1, ... , Xn be iid with the Poisson(λ) distribution. What is the conditional distribution of Xi given the sample mean?
Exercice 6. Let be (Xi,..., Xn) an iid sample from the Bernoulli distribution with parameter θ, ie. I. What is the Maximum Likelihood estimate θ of θ? 2. Show that the maximum likelihood estimator of θ is unbiased. 3. We're looking to cstimate the variance θ (1-9) of Xi . x being the empirical average 2(1-2). Check that T is not unli ator propose an unbiased estimator of θ(1-0).
Let Xi,.., Xn be a sample from Poisson(0) distribution. Consider testing where θ° is a given constant. Use LRT to derive the general form of the union- intersection rejection region in its simplest form.
Let Xi,...,Xn be a random sample from a two parameter exponential distribution with pa- rameter θ (λ, μ), (a) Show that the distribution of Ti = log(X(n)-X) +log λ is free of θ. Îs an ancillary statistics (b) show that 72- Xu is ancillary X-X Let Xi,...,Xn be a random sample from a two parameter exponential distribution with pa- rameter θ (λ, μ), (a) Show that the distribution of Ti = log(X(n)-X) +log λ is free of θ. Îs an...
Let X1,...............,Xn be a sample from an exponential population with parameter λ. (a) Find the maximum likelihood estimator for λ. (b) Is the estimator unbiased? (c) Is the estimator consistent?
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,...
1. Let Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ2-4, where λ is a parameter. Define 2n We use V, to estimate λ. (a) Show that is an unbiased estimator for λ. (b) Let ơin be the variance of V,, . Show that lin ơi,- 1. Let Xi, X2,.., Xn be a random sample drawn from some population with mean μ--2λ and variance σ2-4, where λ is a parameter. Define 2n...
4. Let Xi,X2, , Xn be n i.id. exponential random variables with parameter λ > Let X(i) < X(2) < < X(n) be their order statistics. Define Yǐ = nX(1) and Ya = (n +1 - k)(Xh) Xk-n) for 1 < k Sn. Find the joint probability density function of y, . . . , h. Are they independent? 15In
The number of medical emergency calls per hour has a Poisson distribution with parameter λ. Calls received at different hours are considered to be independent. Emergency calls X1 ,…, Xn for n consecutive hours has the same parameter λ. a) What is the distribution of Sn = ∑ Xi ? b) Provide Normal approximation for the distribution of Sn . c) Provide maximum likelihood estimation of λ. Calculate variance and bias of MLE. d) Calculate Fisher information and efficiency of...
Let X1...Xn be independent, identically distributed random sample from a poisson distribution with mean theta. a. Find the meximum liklihood estimator of theta, thetahat b. find the large sample distribution for (sqrt(n))*(thetahat-theta) c. Construct a large sample confidence interval for P(X=k; theta)