Problem 8.2 Suppose that Xi, X,.., 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...
4. Let Xi,.. . , Xn be a random sample from a distribution with the density function 62(1-x), if 0〈x〈1; f(x) = elswhere. As usual, define First determine the mean and variance of the given distribution. What is an approximate distribution of Xn? For a sample of size 75, what are the exact mean and variance for Xn?
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
valu Exercises 8.2. x, . . . ,x, nd G(p), the geometric distribution with mean 1/p. Assume that e size n is sufficiently large to warrant to invocation of the Central Limit Theo- . Suppose se that Xi , . . . X, Use the asymptotic d confidence interval for p Suppose that XN(0, o2) (a) Obtain the asymptotic distribution of the second istribution of p 1/X to obtain an approximate 100(1-u)% Suppose sample moment m2 -(I/n)i X. (b) Identify...
3. [6 pts] Let Xi, . . . , Xn be a random sample from a distribution with variance σ2 < oo. Find cov(X,-x,x) for i 1,..,n.
3. [6 pts] Let Xi, . . . , Xn be a random sample from a distribution with variance σ2
Let Xi,, Xn be a random sample of size n from the normal distribution with mean parameter 0 and variance σ2-3. (a) Justify thatX X, has a normal distribution with mean parameter 0 and variance 3 /n, this is, X~N(0,3/m) (you can do it formally using m.g.f. or use results from normal distribution to justify (b) Find the 0.975 quantile of a standard normal distribution (you can use a table, software or internet to find the quantile). (c) Find the...
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...
2. Let Xi,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n = 2m+1 for some integer m. Let Y be the sample median and Z = (a) Apply the usual formula for the density of an order statistic to show the density max(X1) be the sample maximum. of Y is 0) 6 3) (b) Note that a beta random variable X has density re+ β22 a-1 (1-2)8-1 with mean μ α/G + β) and variance...
Let X, and Sn be the sample mean and the sample variance of {Xi,.X. Let Xn+1 and S7+1 be the sample mean and the sample variance of {X1,..., Xn, X y. Which of the following hold for sample means, for sample variances? 72 m+1 m+1
(b) Suppose that xi, . . . ,Xn are a random sample of lifetimes for individuals diagnosed with a certain disease. Assume a model with λ, x>0, where k is fixed and known Interest is in the parameter Ç P(X > 25A) which gives the probability that an individual will survive more than 25 years with the disease. It can be shown that the cumulative distribution Determine the MLE of