View Question
 Question
 Subject: Probability - Random Variable* Category: Science > Math Asked by: alpa101473-ga List Price: \$3.50 Posted: 22 Sep 2003 20:18 PDT Expires: 22 Oct 2003 20:18 PDT Question ID: 259269
 ```For question please click on the following link: http://www.ofoto.com/PhotoView.jsp?UV=512437033106_50356301605&collid=12136301605&photoid=91136301605&page=1 If not possible, please leave comment.```
 ```Problem: Let Y = A*cos(w*t) + c, where A has mean m and variance sigma^2, and w and c are constants. Find the mean and variance of Y. In statistics, the mean is often described as the "expectation value", often written as E[X], where X is a statistical variable. (Techicialy, there is a distinction, but for our purposes we can say this; the third link I have provided makes clear the distinction.) In your problem, Y is such a variable. The variance of a varible is commonly notated as V[X]. Looking at some general properties for expectation and variance: if we let a variable Y = aX + b then: E[Y] = a*E[X] + b and: V[Y] = (a^2)*V[X] This is known as a linear transformation on the variable X (multiplying by a constant and adding a constant term). The links I have provided provide proofs of thse properties. Returning to the specific question at hand, we can apply the rules for a linear transformation. Given that Y = A*cos(w*t) + c, we know that E[A] = m and V[A] = sigma^2. Therefore: E[Y] = E[A*cos(w*t) + c] = E[A]*cos(w*t) + c = m*cos(w*t) + c V[Y] = V[A*cos(w*t) + c] = V[A]*(cos(w*t))^2 = (sigma^2)*(cos(w*t))^2 The following sites provide references for basic statistical identities as well as proofs: Iowa State University: Expectation, Mean, Variance http://clue.eng.iastate.edu/~zhengdao/teaching/ee322/spring03/notes/note7/l7.pdf The Aarhus School of Business: Introduction to Expectation and Variance http://www.hha.dk/ifi/QEM/expvar.pdf University of Texas at Dallas: Expectation of functions of random variables http://www.utdallas.edu/~ammann/cs3341/node24.html I hope that answers everything you needed to know. Please request any clarifications you may have. Thank you for using Google Answers. xargon-ga Search used: variance expectation ://www.google.com/search?sourceid=mozclient&ie=utf-8&oe=utf-8&q=variance+expectation```