2CL-S10 Lecture 9

# 2CL-S10 Lecture 9 - Final exam: Wednesday June 2 Location:...

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An 8 ½ x 11 cheat sheet (both sides) will be allowed, recommended, and you can use your calculator during the exam. Final exam : Wednesday June 2 8:00 - 8:50 pm Location: CENTR 101 Exam will consist of 4 problems. 1 optics problem and 3 statistics problems.

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Covariance Consider a function of two variables q ( x , y ) . From N pairs of data ( x i , y i ) we compute N values of q i ( x i , y i ) . Assume that the sets ( x i ) and ( y i ) are close to their mean values and . Then from a Taylor’s series expansion: q i q x i , y i q x , y q x x i x q y y i y . Taking the average of this expression we are led to the simple result: q q x , y . The standard deviation in the N values of q i is: q 2 1 N i 1 N q i q 2 1 N i 1 N q x x i x q y y i y 2 , q 2 q x 2 1 N i 1 N x i x 2 q y 2 1 N i 1 N y i y 2 2 q x q y 1 N i 1 N x i x  y i y . x y
Covariance The sums in the first two terms involve the standard definitions of s x and y . The sum in the third term is called the covariance of x and y : If x and y are uncorrelated (as has been the assumption for earlier results) then for a large number of samples xy tends toward zero! In this case the expression for q reduces to the familiar expression for q . However if the covariance is not zero even in the limit of an arbitrary number of measurements, then x and y are correlated! xy 1 N i 1 N x i x  y i y . It is in this case for which the uncertainty in q must include the term that involves the covariance. q 2 q x 2 x 2 q y 2 y 2 2 q x q y xy .

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Covariance - Example In an effort to get his students to do their homework, a professor wants to make the case that good scores on the exam are correlated to their homework scores. The scores for 10 students are: For these 10 students the mean values for their scores are: The covariance for these scores is: s xy = 367.45. It certainly seems that these scores are correlated as the covariance does not appear to be tending toward zero! x 57.4, y 77.1 Actually it is the size of the covariance when compared to the product x y that is relevant.
Schwartz Inequality The Schwartz inequality states: which is always greater than zero. Since this is true for all t , we want to find the minimum value for A ( t ) which will still be greater than zero.

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## This note was uploaded on 06/04/2010 for the course PHYS 2CL taught by Professor Bodde during the Spring '08 term at UCSD.

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2CL-S10 Lecture 9 - Final exam: Wednesday June 2 Location:...

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