APPLIED STATISTICS M 358K
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This 2 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 358K at University of Texas at Austin taught by Martha Smith in Fall. Since its upload, it has received 89 views. For similar materials see /class/181459/m-358k-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.
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Date Created: 09/06/15
M 358K PROPERTIES OF THE CORRELATION COEFFICIENT You may have seen the covariance of two random variables in M362K CovXY EX EXY EY It is related to the variance by VarX Y VarX VarY 2CovXY This is straightforward to establish from the de nitions You might even have seen the correlation coefficient for random variables 9 CovXYVarXVarY12 We can define the sample covariance of a collection x1 xn yl yn of two variable data as covxy E X YXYF T n l Notice that the sample correlation coefficient defined in the textbook can be expressed as r covxysxsy Also notice that covaxby abcovxy Details left to the student If we use varx to denote the sample variance then we have 1 n varxy quot4211 Xyxy2 1 n EEH xiiyiy2smce xy xy 1 n xi 32 2xi moi5 yiW varx vary 2covxy Now apply this to xsX ysy instead of X y varxsX ysy varxsx varysy 2covxsxysy varxsx2 varysy2 2covxysxsy 2 2r Why Since the sample variance is always 2 0 Why this implies that r 2 1 Now if we had a case with r 1 then we would have varxsX ysy 0 which implies Why Hint Look at the definition of variance that for each i xisX sty x xx y s which we will call c since it is a constant Thus xisX sty c i 1 2 n so yi sy c xisx i 1 2 n In other words the points x1y1 Xn yn all lie on the straight line with slope sysx which is negative Similarly by considering varxsX ysy we can show that r s 1 and that if r 1 then the points x1y1 Xn y all lie on the straight line with slope sysX which is positive Fill in details
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