Let X be a random variable with mean and variance \u03c3 Find the expected value and

Let x be a random variable with mean and variance σ

This preview shows page 32 - 45 out of 45 pages.

Let X be a random variable with mean 𝜇 and variance σ . Find the expected value and variances of random variables 3X 2X+5
Example: Sample Mean & Variance Are Unbiased-1 X is a random variable with mean μ and variance σ 2 . Let X 1 , X 2 ,…, X n be a random sample of size n . Show that the sample mean ( X -bar) is an unbiased estimator of μ .
Show that the sample variance ( S 2 ) is an unbiased estimator of σ 2 .
Minimum Variance Unbiased Estimators If we consider all unbiased estimators of θ , the one with the smallest variance is called the minimum variance unbiased estimator (MVUE). If X 1 , X 2 ,…, X n is a random sample of size n from a normal distribution with mean μ and variance σ 2 , then the sample X - bar is the MVUE for μ . WHY?
Standard Error of an Estimator
Mean Squared Error Conclusion: The mean squared error (MSE) of the estimator is equal to the variance of the estimator plus the bias squared.
Relative Efficiency The MSE is an important criterion for comparing two estimators. If the relative efficiency is less than 1, we conclude that the 1 st estimator is superior than the 2 nd estimator.
Optimal Estimator A biased estimator can be preferred than an unbiased estimator if it has a smaller MSE. Biased estimators are occasionally used in linear regression. An estimator whose MSE is smaller than that of any other estimator is called an optimal estimator. Figure 7-8 A biased estimator that has a smaller variance than the unbiased estimator .
Example-5 Suppose that the random variable X has the continuous uniform distribution Suppose that a random sample of n = 12 observations is selected from this distribution. What is the approximate probability distribution of X − 6 ? Find the mean and variance of this quantity.
Exmple-5
Example-6 A computer software package calculated some numerical summaries of a sample of data. The results are displayed here: (a) Fill in the missing quantities. (b) What is the estimate of the mean of the population from which this sample was drawn?
Example-7 Let X 1 and X 2 be independent random variables with mean μ and variance σ2. Suppose that we have two estimators of μ: (a) Are both estimators unbiased estimators of μ? (b) What is the variance of each estimator?
Example-7

You've reached the end of your free preview.

Want to read all 45 pages?

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes