5.4.5. Let X,, , Xn be a random sample from an Mu,02), where the 19 value...
Let X, , . . ., Xn be a random sample from an N(p, ơ2). (a) Construct a (1-α) 100% confidence interval for μ when the value of σ2 is known. (b) Construct a (1-α) 100% confidence interval for μ when the value of σ2 is unknown.
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n). Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
2) (1 point) Let Xi, , Xn be a random sample from V(0, σ2). And their corresponding observations are x1, . . . , Xn. Show that is a (1-a) 100% confidence interval of σ2. 2) (1 point) Let Xi, , Xn be a random sample from V(0, σ2). And their corresponding observations are x1, . . . , Xn. Show that is a (1-a) 100% confidence interval of σ2.
Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a 100(1-a)% confidence interval for o2 based on the sample variance S2. Leave your answer in terms of chi-squared critical values. (Hint: We will show in class that, for this normal sample, (n − 1)S2/02 ~ x?(n − 1).)
Let X1, ..., Xn be a random sample from the distribution 1 f(x; 01, 02) e-(2–01)/02 x > 01, - < 01 <0, 02 > 0. 7 02 Find the method of moments estimators (MMEs) of 04 and 02.
1.Suppose X1, X2, .., Xn is a random sample from N(", 02) 10 pts] If o2 1, u is unknown. Find the MLE of a. b. [10 pts If o2 = 1, p is unknown. f = X is an estimator of u. What is the MSE of this estimator? Now assume o2 is unknown. The following data is a set of observations of X1,..., Xn. Use the dataset to answer (c), (d) and (e) 11 8 9 7 6...
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 X1, ..., Xn be a random sample (i.i.d.) from a normal distribution with parameters µ, σ2 . (a) Find the maximum likelihood estimation of µ and σ 2 . (b) Compare your mle of µ and σ 2 with sample mean and sample variance. Are they the same?
Let X1, X2, ...,Xn be a random sample of size n from a Poisson distribution with mean 2. Consider a1 = *1782 and în = X. Find RE(21, 22) for n = 25 and interpret the meaning of the RE in the context of this question.
Let X1,... , Xn be a random sample from the Pareto distribution with pdf Ox (0+1), x > 1, f(z0) where 0>0 is unknown (a) Find a confidence interval for 0 with confidence coefficient 1-a by pivoting a ran- dom variable based on T = T log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval (b) Find a confidence interval for 0 with confidence coefficient 1 - a by pivoting the cdf...