Corelation Coefficient

Definition (Corelation Coefficient).

The correlation coefficient between and Y

Lemma.

let standard deviation of

similarly

Remark.

Say with Probability 1
if

Remark.

Recall for discrete Random variable, we need to focus on values that RV takes
that is with for all

is measure of linear relation between and
if high degree of linearity
say , is uncorrelated

Example

independent independent and identically identicallydistribution (IID)
consider independent trials each of which results in any of possible outcomes with Probability

if
lets take

Verify the above result as completed it