probability :- covariance and correlation [email protected]
DESCRIPTION
it helps to understand the topicTRANSCRIPT
1Jerrell T.Stracener – Ph.D.
Covariance & Correlation
Covariance – Cov(X,Y)
Covariance between X and Y is a measure of the association between two random variables, X & Y If positive, then both move up or down together If negative, then if X is high, Y is low, vice versa
YXXY YXEYXCov ),(
Correlation Between X and Y
Covariance is dependent upon the units of X & Y [Cov(aX,bY)=abCov(X,Y)] Correlation, Corr(X,Y), scales covariance by the standard deviations of X & Y so that it lies between 1 & –1
2
1
)()(
),(
YVarXVar
YXCov
YX
XYXY
More Correlation & Covariance
If X,Y =0 (or equivalently X,Y =0) then X and Y are linearly unrelated
If X,Y = 1 then X and Y are said to be perfectly positively correlated
If X,Y = – 1 then X and Y are said to be perfectly negatively correlated Corr(aX,bY) = Corr(X,Y) if ab>0 Corr(aX,bY) = –Corr(X,Y) if ab<0
Properties of Expectations
E(a)=a, Var(a)=0
E(X)=X, i.e. E(E(X))=E(X)
E(aX+b)=aE(X)+b
E(X+Y)=E(X)+E(Y)
E(X-Y)=E(X)-E(Y)
E(X- X)=0 or E(X-E(X))=0
E((aX)2)=a2E(X2)
More Properties
Var(X) = E(X2) – x2
Var(aX+b) = a2Var(X)
Var(X+Y) = Var(X) +Var(Y) +2Cov(X,Y)
Var(X-Y) = Var(X) +Var(Y) - 2Cov(X,Y)
Cov(X,Y) = E(XY)-xy
If (and only if) X,Y independent, then Var(X+Y)=Var(X)+Var(Y), E(XY)=E(X)E(Y)
7Jerrell T.Stracener – Ph.D.
Covariance of X and Y
Let X and Y be random variables with joint mass function p(x,y) if X & Y are discrete random variables or with joint probability density function f(x, y) if X & Y are continuous random variables. The covariance of X and Y is
if X and Y are discrete, and
if X and Y are continuous.
yxpyxYXEx y
yxYXXY ,
dxdyyxfyxYXE yxYXXY ,
8Jerrell T.Stracener – Ph.D.
Covariance of X and Y
The covariance of two random variables X and Y with means X and Y , respectively is given by
YXXY XYE
9Jerrell T.Stracener – Ph.D.
Correlation Coefficient
Let X and Y be random variables with covariance XY and standard deviation X and Y , respectively. The correlation coefficient of X and Y is
YX
XYXY
10Jerrell T.Stracener – Ph.D.
Theorem
If X and Y are random variables with joint probability distribution f(x, y), then
XYYXbYaX abba 222222
11Jerrell T.Stracener – Ph.D.
Theorem
If X and Y are independent random variables, then
22222YXbYaX ba
12Jerrell T.Stracener – Ph.D.
Correlation Analysis
A statistical analysis used to obtain a quantitative measure of
the strength of the linear relationship between a dependent
variable and one or more independent variables
13Jerrell T.Stracener – Ph.D.
Correlation – Scatter Diagram
Visual Relationship Between X and Y