1. A density function sometimes used by engineers to model length of life of electronic components is the Rayleigh density, given by if 0 < y1 <infinity, 0 otherwise. Assume Y1, Y2, ...Yn is a random sample from a Rayleigh distribution.
a. If Y has the Rayleigh density, find the probability density for U = Y2.
b. Use part a to find E(Y) and V(Y).
c. Use the first moment E(Y) to find a method of moment estimator for .
d. Use the second moment E(Y) to find a method of moment estimator for .
e. Find the MLE of . Denote this as .
f. What is the asymptotic variance of ?
g. Show whether is an unbiased estimator for .
h. Show whether is a consistent estimator for .
i. Show whether is sufficient for .
The moment and maximum likelihood estimation of the parameters of a Rayleigh density are discussed in the solution. The examination of unbiasedness, consistency, sufficiency of the proposed estimators are also discussed.
Defining statistics terms: Sampling error, estimators, central limit theorem, standard error
Define (a) parameter, (b) estimator, (c) sampling error, and (d) sampling distribution.
Explain the difference between sampling error and bias. Can they be controlled?
Name three estimators. Which ones are unbiased?
Explain what it means to say an estimator is (a) unbiased, (b) efficient, and (c) consistent.
State the main points of the Central Limit Theorem for a mean.
Why is population shape of concern when estimating a mean? What does sample size have to do
(a) Define the standard error of the mean. (b) What happens to the standard error as sample size
increases? (c) How does the law of diminishing returns apply to the standard error?
Define (a) point estimate, (b) interval estimate, (c) confidence interval, and (d) confidence level.
List some common confidence levels. Why not use other confidence levels?
(a) List the steps in testing a hypothesis. (b) Why can't a hypothesis ever be proven?
(a) Explain the difference between the null hypothesis and the alternative hypothesis. (b) How is the
null hypothesis chosen (why is it "null")?
(a) Why do we say "fail to reject H0" instead of "accept H0"? (b) What does it mean to "provisionally
accept a hypothesis"?
(a) Define Type I error and Type II error. (b) Give an original example to illustrate.
(a) Explain the difference between a left-sided test, two-sided test, and right-sided test. (b) When
would we choose a two-sided test? (c) How can we tell the direction of the test by looking at a pair