Covariance and Correlation Properties

T/F: if X and Y are independent, then correlation(X,Y) = 0

True

cov(X,X) = ?

Var(X)

correlation(X,Y) = 1 <=> ? correlation(X,Y) = -1 <=> ?

Y = mx+b with m > 0 (positive slope) Y = mx+b with m < 0 (negative slope)

cov(αX, βY) = ?

αβcov(X,Y)

cov(X,α) = ?

0

T/F: A correlation of 0 means that there's no relationship between two variables

False - it doesn't mean there's NO relationship, it just means there's NO LINEAR relationship

T/F: if E[XY] = E[X]E[Y], then X and Y are independent

False - see notes for counter example of x and y = x²

T/F: if correlation(x,y) = 0, then x and y are independent

False

Independence vs Statistical Independence

* Independence: joint density is factorable and defined on a rectangle like set * Statistical Independence: covariance / correlation = 0

? ≤ correlation(X,Y) ≤ ?

-1 ≤ correlation(X,Y) ≤ 1

Definition of Correlation

Let X and Y be random variables. Zx = X - E[X] / σx Zy = Y - E[Y] / σy Correlation = cov(Zx, Zy) - it is the *linear* measure of dependence

Units of correlation

correlation is *unit-free* -> has no units

correlation(X,Y) = ? -> linear transformation

correlation(ax+b, cy+d) where a > 0, c >0, b∈R, d∈R or a < 0, c <0, b∈R, d∈R

Covariance/correlation when X and Y are independent

cov(X, Y) = 0 (i.e. E[XY] = E[X]E[Y]) correlation = 0

Definition of covariance

cov(X,Y) = E[XY] - E[X]E[Y] where E[XY] = ∫∫xyf(x,y)dxdy

cov(X+Y, Z) = ?

cov(X,Z) + cov(Y,Z)

cov(X,Y) = ?

cov(Y, X)

correlation measures only _________ dependence between X and Y

only *linear* dependence - If the dependence is non-linear then rho is an inapplicable/inappropriate measure