midterm3practice

# midterm3practice - Practice problems for Final Exam Problem...

This preview shows pages 1–2. Sign up to view the full content.

Practice problems for Final Exam Problem 1 A sample X 1 , X 2 , . . . , X n comes from a continuous dis- tribution with the density f X ( x ) = ( θ + 1) x θ , 0 < x < 1 for some unknown parameter θ > - 1. ( a ) Compute the maximum likelihood estimator of θ . ( b ) Compute the moment estimator of θ . ( c ) Compute the Bayes estimator of θ that maximizes the posterior density if the prior density of θ is p ( θ ) = e - ( θ +1) for θ > - 1. Problem 2 Suppose that the daily maximal air temperatures X 1 and X 2 , observed on two successive days follow a bivariate normal distribution with parameters μ 1 = μ 2 = 75, σ 1 = σ 2 = 8 and ρ = . 9. ( a ) Find the probability that the average over two successive days of the daily maximal air temperatures exceeds 80. ( b ) Find the probability that the maximal air temperature on the second day exceeds 80 given that the maximal air temperature on the first day was equal to 80. Problem 3 Suppose that X has the uniform distribution in the interval [0 , T ], but T itself is random, and can take values 1 and 2 with probabilities 1 / 2 each. ( a ) Compute the mean and the variance of X . ( b ) Find the probability P ( X > 1). Problem 4 A sample A sample X 1 , . . . , X n comes from a continuous distribution with the density f X ( x ) = θ ( θ - 1) x (1 + x ) θ +1 , x > 0 for some unknown parameter θ > 1.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### 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