Lecture 6. G) Some Inequalities
Cauchy-Schwarz Inequality
If [math]\left(X,Y\right)'[/math] is a bivariate random vector, then
[math]\left|E\left(XY\right)\right|\leq E\left(\left|XY\right|\right)\leq\sqrt{E\left(X^{2}\right)}\sqrt{E\left(Y^{2}\right)}[/math].
The inequality binds joint moments by their separate properties.
Generalization: Holder Inequality
If [math]\left(X,Y\right)'[/math] is a bivariate random vector, then
[math]\left|E\left(XY\right)\right|\leq E\left(\left|XY\right|\right)\leq\sqrt[p]{E\left(\left|X\right|^{p}\right)}\sqrt[q]{E\left(\left|Y\right|^{q}\right)},\,for\,p,q\gt 0,p^{-1}+q^{-1}=1.[/math]
Jensen’s Inequality
If [math]X[/math] is an r.v. and [math]g:\mathbb{R}\rightarrow\mathbb{R}[/math] is a convex function, then
[math]E\left[g\left(X\right)\right]\geq g\left[E\left(X\right)\right].[/math]
This also implies that if [math]g\left(\cdot\right)[/math] is concave, then
[math]E\left[g\left(X\right)\right]\leq g\left[E\left(X\right)\right].[/math]
Finally, if [math]g\left(\cdot\right)[/math] is linear, then
[math]E\left[g\left(X\right)\right]=g\left[E\left(X\right)\right],[/math]
since a linear function is both convex and concave.
For an example, consider an r.v. [math]X[/math] that can equal [math]0[/math] or [math]8[/math] with equal probability.
In this case,
[math]E\left(X\right)^{2}=\left(0.5\times0+0.5\times8\right)^{2}=16[/math]
and
[math]E\left(X^{2}\right)=\left(0.5\times0^{2}+0.5\times8^{2}\right)=32,[/math]
as predicted by Jensen's inequality, since [math]y=x^{2}[/math] is convex.
Consider a graphical depiction is below:
The curves above are obtained by varying the probability weights associated with 0 and 8 from zero to one. Jensen's inequality applies throughout the graph.
