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Unformatted text preview: E éêë(X - mX )(Y - mY )ùúû VER. 9/11/2012. © P. KOLM 17 Correlation – Corr(X,Y) Covariance is dependent upon the units of X and Y (i.e.
Cov(aX ,bY ) = a ⋅ b ⋅ Cov(X ,Y ) ) Correlation, denoted by Corr (X ,Y ) or rXY , “normalizes” the covariance by the
standard deviations of X and Y,
Corr (X ,Y ) º rXY º sXY
Cov(X ,Y )
Var (X ) ⋅Var (Y ) We therefore have that -1 £ Corr (X ,Y ) £ 1 VER. 9/11/2012. © P. KOLM 18 More on Correlation and Covariance If sXY = 0 (or equivalently rXY = 0 ) then X and Y are linearly unrelated
If rXY = 1 then X and Y are said to be perfectly positively correlated
If rXY = -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 VER. 9/11/2012. © P. KOLM 19 Some Properties of the Expected Value Let X and Y, and a and b, denote two random variables and two constants,
respectively. Then, E (a ) = a , Var (a ) = 0 E (E (X )) = E (X ) , i.e. E (mX ) = mX E (aX + b) = aE (X ) + b E (X +Y ) = E (X ) + E (Y ) E ((aX )2 ) = a 2E (X 2 ) VER. 9/11/2012. © P. KOLM 20 Some Properties of the Variance and the Covariance
2 Var (X ) = E (X 2 ) - mX Var (aX + b) = a 2Var (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 ) - mX mY If X and Y are independent, then Cov(X ,Y ) = 0 . (Note: The converse is not true.) (For you: Verify these properties.) VER. 9/11/2012. © P. KOLM 21 The Normal Distribution We denote by N (m, s 2 ) the normal distribution, with mean m and variance s 2 The probability density function (pdf) of the normal distribution is given by
æ (x - m)2 ö
f (x ) =
s 2p VER. 9/11/2012. © P. KOLM 22 The Standard Normal Distribution Any random variable can be “standardized” by (1) subtracting its mean, and (2) dividing by its standard deviation. In other words, by “standardizing” X we X - mX
obtain the new random variable i.e. Z =
o Note that E (Z ) = 0 and Var (Z ) = 1 A standardized normal variable follows a standard normal distribution, denoted by N (0,1) . Its pdf is given by j(z ) = VER. 9/11/2012. © P. KOLM 1
2p æ -z 2 ö
ø 23 Some Properties of the Normal If X N (m, s 2 ) then aX + b N (a m + b, a 2s 2 ) A linear combination of independent, identically distributed (iid) normal random variables are normally distributed
s2 If Y1,Y2,... n N (m, s ) are iid then E (Y ) N (m, )
2 VER. 9/11/2012. © P. KOLM 24 The Cumulative Distribution Function For a pdf, f (x ), where f (x ) P(X = x ), the cumulative distribution function (cdf), F (x ) , is P(X £ x ) ; P (X > x ) = 1 - F (x ) = P (X < -x ) The pdf for the standard normal is f(z ) . Its cdf is denoted by F(z ) = P (Z < z )
o P ( Z > a ) = 2P (Z > a ) = 2[1 - F(a )]
o P (a £ Z £ b) = F(b) - F(a ) VER. 9/11/2012. © P. KOLM 25 The Chi-Square Distribution The chi-square distribution is obtained directly from independ...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
- Fall '14