In this segment, we justify some of the property is that
the correlation coefficient that we
claimed a little earlier.
The most important properties of the correlation coefficient
lies between minus 1 and plus 1.
We will prove this property for the special case where we
have random variables with zero means and unit variances.
So standard deviations are also 1, so most of the terms
here disappear and the correlation coefficient is
simply the expected value of X times Y.
We will show that in this special case the expected
value of X times Y lies between minus 1 and 1.
But the proof of this fact remains valid with a little
bit of more algebra along similar lines
for the general case.
What we will do is we will consider this quantity here
and expand this quadratic and write it as
expected value of X squared.
Then there's a cross term, which is minus 2 rho, the
expected value of X times Y, plus rho squared, expected
value of Y squared.
Now since we assume that the random variables have 0 mean,
this is the same as the variance and we assume that
the variance is 1, so this term here is equal to 1.
Now, the expected value of X times Y is the same as the
correlation coefficient in this case.
So we have minus 2 rho squared and from
here we have rho squared.
And by the previous argument, again this quantity, according
to our assumptions, is equal to 1 so we're left with this
expression, which is 1 minus rho squared.
Now, notice that this is the expectation of a non-negative
random variable so this quantity here must be
non-negative.
Therefore, 1 minus rho squared is non-negative, which means
that rho squared is less than or equal to 1.
And that's the same as requiring that rho lie between
minus 1 and plus 1.
And so we have established this important property, at
least for the special case of 0 means and unit variances.
But as I mentioned, it remains valid more generally.
Now let us look at an extreme case, when the absolute value
of rho is equal to 1.
What happens in this case?
In that case, this term is 0 and this implies that the
expected value of the square of this random variable is
equal to 0.
Now here we have a non-negative random variable,
and its expected value is 0, which means that when we
calculate the expected value of this there will be no
positive contributions and so the only contributions must be
equal to 0.
This means that X minus rho Y has to be equal to 0 with
probability 1.
So X is going to be equal to rho times Y and this will
happen with essential certainty.
Now also because the absolute value overall is equal to 1,
this means that we have either X equal to Y or X equals to
minus Y, in case rho is equal to minus 1.
So we see that if the correlation coefficient has an
absolute value of 1, then X and Y are related to each
other according to a simple linear relation, and it's an
extreme form of dependence between
the two random variables.