Cauchy-Schwarz inequality

Cauchy-Schwarz inequality for random variables

 The value of the correlation coefficient of two random variables X and Y is a number between -1 and + 1. This result is a consequence of the Cauchy-Schwarz inequality which states that :

When the inequality is in fact an equality, we'll show that X and Y are then linearly related : there exist two numbers a and b such that :

Y = aX + b

-----

The Cauchy-Schwarz inequality is to be found in many places Statistics. For example, it is a key element in establishing the Cramér-Rao lower bound.

Scope of the Cauchy-Schwarz inequality

The scope of the Cauchy-Schwarz inequality extends way beyond the bounds of Statistics. It is a consequence of defining the inner product of two vectors in vector spaces, and is therefore to be found in many areas throughout Mathematics.

It is easily interpreted when applied in elementary Geometry. This illustration shows two vectors V₁ and V₂ :

The Cauchy-Schwarz inequality states that the square of the inner product V₁.V₂ is less than the product of the squared lengths of V₁ and of V₂.

But :

V₁.V₂  =  | V₁ |.| V₂ |.cosα

and the Cauchy-Schwarz inequality therefore states that :

-1 cosα  +1

 _________________________________________________________________

In this short Tutorial, we demonstrate two special versions of the Cauchy-Schwarz inequality :

    * The first one pertains to random variables (see above).

    * The second one pertains to functions that are both integrable and whoses squares are also integrable.

CAUCHY-SCHWARZ INEQUALITY

__________________________________________________

Related readings :