Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
AMS 311, Lecture 23 May 3, 2001 Final Examination Next Thursday (May 10), 11 am to 1:30 in this classroom. The examination is comprehensive. You may use one reference document and your calculator. See exam week office hours give in Lecture 22 handout and posted on web site. Review: Linear combinations of random variables can be written in vector notation as W a T Y . Then E (W ) a T E (Y ), and var(W ) a T vcv(Y )a. Moment generating functions. Practical Problems on the Central Limit Theorem The winnings W in a game of chance have an expected value $25 and variance 1,000,000. Describe the distribution of S100 100 W , the total winnings after 100 independent plays i 1 i of the game of chance. What is the value of the reserve r that you should hold so that P( S100 r ) 0.01? Statistical applications make frequent use of the central limit theorem to specify a test and to find the properties of a test. Conditioning on random variables. Recall that E ( X |Y ) (Y ) is a random variable (called the regression function). There are two fundamental identities: E ( E ( X |Y )) E ( X ). This is deceptively easily stated. Make sure that you understand the probability measure governing each expectation. Similarly, var( X |Y ) is also a random variable (it is also a function of Y).The second fundamental identity is var( X ) E (var( X |Y )) var( E ( X |Y )). The second fundamental identity is reflected in the basic identity of the statistical analysis of linear models: Total sum of squares=sum of squares due to model and sum of squares due to error. Chebyshev’s Inequality If X is a random variable with expected value and variance 2, then for any t>0, P(| X | t ) 2 t2 .