Let X ~ N(μ = 5, σ2-22). In what follows, use pnorm() to compute probabilities. a)...
, X,' up N(μ, σ2), with σ2 known. Let μη-Xn + 5. Let Xi, of u be an estimator (a) Is ,hi an unbiased estimator for μ? (b) For a particular fixed n, find the distribution of (c) Find the mean squared error (MSE) of . (d) Prove that μη is consistent for μ
Please explain very carefully!
4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 > 0 are unknown. (a) (5 marks) Let μ+σ~p denote the p-th quantile of the N(μ, σ*) distribution. What does this mean? (b) (10 marks) Determine a UMVU estimate of,1+ ơZp and justify your answer.
4. Suppose that x = (x1, r.) is a sample from a N(μ, σ2) distribution where μ E R, σ2 >...
X follows normal distribution N (μ, σ2) with pdf f and cdf F. If max, f (x)-0.997356 and F (-1) + F (7-1, determine P(X s 0)
Let X equal the weight of the soap contained in a box. Assume that the distribution of X is N(μ = 6.05, σ2 = 0.0004). (a) Compute the probability P(X < 6.0171) (b) A random sample of nine (9) boxes of soap is selected from the production line. Let Y equal the number of boxes that weigh less than 6.0171. What is the distribution of Y? (c) Find the probability that at most two (2) boxes weigh less than 6.0171. (d) Let X̅ be...
Exercice 5. Let Xi, ,Xn be iid normal randon variables : Xi ~ N(μ, σ2). We denote 4 Tl Show that (İ) ils2 (i.e., that x is independent of 82). (ii) x ~ N(μ, σ2/n). (iii) !뷰 ~ เลี้-1
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
X follows normal distribution N (μ, σ2) with pdf f and cdf F. if max, f (z) = 0.997356 and F (-1) + F (7)-1, determine .4 .4
Find μ if μ ΣΙΧ.P(x)]. Then, find σ if σ2 ΣΙΧ2 . P(x)-μ2. 2 5 2 P)0.0002 0.00530.0450 0.19190.4089 0.3487 H(Simplify your answer. Round to four decimal places as needed.) ơ- (Simplify your answer. Round to four decimal places as needed.) Enter your answer in each of the answer boxes.
5. Suppose X. N(μ, σ2), what is the distribution of the sample mean Σ ? Comment on the behavior of the distribution for increasing n. Furthermore, is the distribution of the sample mean consistent with the predictions of the central limit theorem?
Exercises 10.3. Let Xi . . . , x N μ, σ2), whereơ2 s known to be equal to 100. In testing Ho : 25vs. H :H>25,h What sample size n would be necessary if one wishes to reject Ho with probability at least 95 if μ 26? iid se that a coin is to be tossed n times, and you wish to test the hypothesis Ho:p-12 VS. Hi P> I/2 at a- .05. What sample size n would be...