correlation

Correlation
The correlation of two random variables X and Y , denoted by ?(X, Y ), is defined by
?(X, Y ) = cov(x, y)/?X ?Y
If ?X = 0 or If ?Y = 0 then ?(X, Y ) = 0.
The random variables X and Y are said to be uncorrelated if and only if
?(X, Y ) = 0; otherwise they are said to be correlated.
Note that if X and Y are independent random variables, then they are uncorrelated. But the opposite is not true (i.e. we can find two random variable that are uncorrelated but not independent).
Finally, it is worth noting the following properties of ?:
?(X, Y ) = ?(Y, X),
?1 ? ? ? 1,
?(X, X) = 1, ?(X, ?X) = ?1.

Example:-
Let the joint density function
f(x,y)=8xy ,0?x?1 ,,0?y?x
Find the correlation of x and y.
Sol:-
Since cov(x,y)=4/225
To find correlation ,we first need to find varaincees of x an of y.
E(x^2 )=?_0^1??4x^5 ? dx=2/3, E(y^2 )=?_0^1??4y^3 (1-y^2 ) ? dy=1/3
Thus var(x)=2/3 –(4/5)2 =2/75
var(y)=1/3-(8/15)2=11/225
?(X, Y ) = cov(X, Y)/?X ?Y
( 4/225)/(2/75)( 11/225)=4/?66.