practice-solution

# practice-solution - Mathematics 136, Midterm 2007 Write...

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

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.

Unformatted text preview: Mathematics 136, Midterm 2007 Write your name and sign the Honor code in the blue books provided. This is a closed material exam, but you may use three pages (2 sides each) of notes. You have 90 minutes to solve all three questions, each worth points as marked (maximum of 50). Complete reasoning is required for full credit. You may cite lecture notes and homework sets, as needed, stating precisely the result you use, why and how it applies. 1. (4x4) Provide the complete, accurate and rigorous definition of four of the following concepts. a) A random variable X defined on ( , F , P ) and its law P X . ANS: Detail Definitions 1.2.1 and 1.4.1. b) The space L 2 ( , F , P ) and the convergence in 2-mean of random variables X n to a random variable X . ANS: Detail the introduction to Section 1.3.2 and Definition 1.3.17. c) The conditional expectation of X L 1 ( , F , P ) given a -field G F . ANS: Detail Definition 2.1.5. d) The characteristic function of a random vector X with values in IR n , and a Gaussian random vector Y with values in IR n . ANS: Detail Definitions 3.2.1 and 3.2.8. e) A version of a stochastic process and a modification of a stochastic process. ANS: Detail Definitions 3.1.8 and 3.1.9. 2. (4+2+2+2+2+4) Let = (0 , 1] with the corresponding Borel -field B and the (uniform) probability measure U on this measurable space....
View Full Document

## practice-solution - Mathematics 136, Midterm 2007 Write...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online