Sample Correlation. Mathematics 47: Lecture 5. Dan Sloughter. Furman University. March 10, 2006
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1 Sample Correlatio Mathematics 47: Lecture 5 Da Sloughter Furma Uiversity March 10, 2006 Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
2 Defiitio If X ad Y are radom variables with meas µ X ad µ Y ad variaces σx 2 ad σy 2, respectively, the we call the covariace of X ad Y. cov(x, Y ) = E[(X µ X )(Y µ Y )] Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
3 Theorem (Cauchy-Schwarz Iequality) If X ad Y are radom variables for which E[X 2 ] ad E[Y 2 ] both exist, the (E[XY ]) 2 E[X 2 ]E[Y 2 ]. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
4 Theorem (Cauchy-Schwarz Iequality) If X ad Y are radom variables for which E[X 2 ] ad E[Y 2 ] both exist, the (E[XY ]) 2 E[X 2 ]E[Y 2 ]. Proof. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
5 Theorem (Cauchy-Schwarz Iequality) If X ad Y are radom variables for which E[X 2 ] ad E[Y 2 ] both exist, the (E[XY ]) 2 E[X 2 ]E[Y 2 ]. Proof. Let f (t) = E[(X + ty ) 2 ] = E[X 2 ] + 2tE[XY ] + t 2 E[Y 2 ]. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
6 Theorem (Cauchy-Schwarz Iequality) If X ad Y are radom variables for which E[X 2 ] ad E[Y 2 ] both exist, the (E[XY ]) 2 E[X 2 ]E[Y 2 ]. Proof. Let f (t) = E[(X + ty ) 2 ] = E[X 2 ] + 2tE[XY ] + t 2 E[Y 2 ]. The f is a quadratic polyomial i t with f (t) 0 for all t. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
7 Theorem (Cauchy-Schwarz Iequality) If X ad Y are radom variables for which E[X 2 ] ad E[Y 2 ] both exist, the (E[XY ]) 2 E[X 2 ]E[Y 2 ]. Proof. Let f (t) = E[(X + ty ) 2 ] = E[X 2 ] + 2tE[XY ] + t 2 E[Y 2 ]. The f is a quadratic polyomial i t with f (t) 0 for all t. Hece, by the quadratic formula, 4(E[XY ]) 2 4E[X 2 ]E[Y 2 ] 0. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
8 Theorem (Cauchy-Schwarz Iequality) If X ad Y are radom variables for which E[X 2 ] ad E[Y 2 ] both exist, the (E[XY ]) 2 E[X 2 ]E[Y 2 ]. Proof. Let f (t) = E[(X + ty ) 2 ] = E[X 2 ] + 2tE[XY ] + t 2 E[Y 2 ]. The f is a quadratic polyomial i t with f (t) 0 for all t. Hece, by the quadratic formula, 4(E[XY ]) 2 4E[X 2 ]E[Y 2 ] 0. Hece (E[XY ]) 2 E[X 2 ]E[Y 2 ]. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
9 Correlatio coefficiet Applyig the Cauchy-Schwarz iequality to the defiitio of covariace, we have cov(x, Y ) E[(X µ X ) 2 ] E[(Y µ Y ) 2 ] = σ X σ Y. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
10 Correlatio coefficiet Applyig the Cauchy-Schwarz iequality to the defiitio of covariace, we have cov(x, Y ) E[(X µ X ) 2 ] E[(Y µ Y ) 2 ] = σ X σ Y. If we let, the 1 ρ X,Y 1. ρ X,Y = cov(x, Y ) σ X σ Y Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
11 Correlatio coefficiet Applyig the Cauchy-Schwarz iequality to the defiitio of covariace, we have cov(x, Y ) E[(X µ X ) 2 ] E[(Y µ Y ) 2 ] = σ X σ Y. If we let, the 1 ρ X,Y 1. ρ X,Y = cov(x, Y ) σ X σ Y Moreover, ρ X,Y = 1 if ad oly if Y = ax + b for some real umbers a ad b. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
12 Correlatio coefficiet Applyig the Cauchy-Schwarz iequality to the defiitio of covariace, we have cov(x, Y ) E[(X µ X ) 2 ] E[(Y µ Y ) 2 ] = σ X σ Y. If we let, the 1 ρ X,Y 1. ρ X,Y = cov(x, Y ) σ X σ Y Moreover, ρ X,Y = 1 if ad oly if Y = ax + b for some real umbers a ad b. Defiitio We call ρ X,Y the correlatio coefficiet of X ad Y. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
13 Correlatio ad idepedece Note: if X ad Y are idepedet, the cov(x, Y ) = 0 (ad hece ρ X,Y = 0). Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
14 Correlatio ad idepedece Note: if X ad Y are idepedet, the cov(x, Y ) = 0 (ad hece ρ X,Y = 0). If cov(x, Y ) = 0, we say X ad Y are ucorrelated. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
15 Correlatio ad idepedece Note: if X ad Y are idepedet, the cov(x, Y ) = 0 (ad hece ρ X,Y = 0). If cov(x, Y ) = 0, we say X ad Y are ucorrelated. However, ucorrelated does ot ecessarily imply idepedece, although it does if (X, Y ) has a bivariate ormal distributio. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
16 Sample correlatio Now suppose (X 1, Y 1 ), (X 2, Y 2 ),..., (X, Y ) are idepedet idetically distributed pairs of radom variables (that is, a radom sample from a bivariate distributio). Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
17 Sample correlatio Now suppose (X 1, Y 1 ), (X 2, Y 2 ),..., (X, Y ) are idepedet idetically distributed pairs of radom variables (that is, a radom sample from a bivariate distributio). Let R = = 1 1 (X i X )(Y i Ȳ ) (X i X ) 2 1 (Y i Ȳ ) 2 X iy i X Ȳ X i 2 X 2 Y i 2 Ȳ 2 = X 2 i P Y i P X iy i X i (P X i) 2 i Yi 2. (P Y i) 2 Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
18 Sample correlatio Now suppose (X 1, Y 1 ), (X 2, Y 2 ),..., (X, Y ) are idepedet idetically distributed pairs of radom variables (that is, a radom sample from a bivariate distributio). Let R = = 1 1 (X i X )(Y i Ȳ ) (X i X ) 2 1 (Y i Ȳ ) 2 X iy i X Ȳ X i 2 X 2 Y i 2 Ȳ 2 = X 2 i P Y i P X iy i X i (P X i) 2 i Yi 2. (P Y i) 2 Defiitio We call R the sample correlatio coefficiet. Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
19 Example Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
20 Example A experimet to measure the yield of wheat for seve differet levels of itroge gave the followig observatios: Nitroge/acre (x) Yield (cwt/acre) (y) Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
21 Example A experimet to measure the yield of wheat for seve differet levels of itroge gave the followig observatios: Nitroge/acre (x) Yield (cwt/acre) (y) If we let x i ad y i, i = 1, 2,..., 7, represet the itroge levels ad wheat yields, respectively, the 7 x i = 700, 7 y i = 171.3, 7 x i y i = 18, 624, 7 x 2 i = 81, 200, ad 7 y 2 i = Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
22 Example A experimet to measure the yield of wheat for seve differet levels of itroge gave the followig observatios: Nitroge/acre (x) Yield (cwt/acre) (y) If we let x i ad y i, i = 1, 2,..., 7, represet the itroge levels ad wheat yields, respectively, the 7 x i = 700, 7 y i = 171.3, 7 x i y i = 18, 624, So r = 7 x 2 i = 81, 200, ad 7 18, 624 (700)(171.3) 81, 200 (700)2 7 7 y 2 i = (171.3) Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
23 Example (cot d) If the itroge levels are i a vector x ad the wheat yields are i a vector y, the the R commad > cor(x, y) returs r, i this case Da Sloughter (Furma Uiversity) Sample Correlatio March 10, / 8
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