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