{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Stat312: Midterm I Solution Moo K. Chung, Yulin Zhang [email protected], [email protected] March 5, 2003 1. Let X 1 , · · · , X n be a random sample from Bernoulli distribution with parameter p . (a) What is ( S 2 /p 2 ) ? S 2 is the sample variance. Ex- plain your results (10 points). (b) Find an unbiased estimator of p 2 . Explain your results (5 points). Solution. (a) The sample variance is an unbiased esti- mator of the population variance. Hence ( S 2 /p 2 ) = ( S 2 ) /p 2 = Var ( X i ) /p 2 . The variance for a Bernoulli random variable can be esily computed as Var ( X i ) = p (1 - p ) . So ( S 2 /p 2 ) = (1 - p ) /p . (b) We know S 2 and ¯ X will be unbiased estimators of population variance p (1 - p ) and mean p respecively. So E ( S 2 ) - E ( ¯ X ) = p (1 - p ) - p = - p 2 . Hence, ¯ X - S 2 is an unbiased estimator of p 2 . 2. Let X 1 , X 2 be a random sample from N (0 , 1 ) . Note that the sample size is 2 and the density function for X i is f ( x i ) = θ 2 π exp( - θx 2 i / 2) . (a) Obtain an estimator of θ using the method of mo- ments. Explain your results (5 points).
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern