miscellaneous

After years' study in probability, I felt more and more happy with the construction of probability space, it's so simple seemingly but interesting that may be applied to many more areas rather than math.

Let’s review a little bit about the probability space in math. Given a space $(\Omega, \mathbb{F}, \mathbb{P})$, assume that there exists a r.v $X s.t. \{\omega|X(\omega)>0\}\in\mathbb{F}$, then we can measure the events $\{X>0\}$ by using $\mathbb{P}(X>0)$. What’s more, adding a time dimension to make it a stochastic process $X_t(\omega)$, then we have properties like $\mathbb{F}_t\subset\mathbb{F}_{t^+}$ and so on.

Secondly, let’s consider another fact, different people have different decisions when given tasks, why is that? That apparently depends on the knowledge we own, how do I relate this decision making base on probability space construction idea? This attracts me these days and I’ll post what I think later.

To be continued……

A comment on a simple proof of Cauchy inequality

In mathematics, the inequality

$n({{x}_{1}}^{2}+\cdots +{{x}_{n}}^{2})\geq {{({{x}_{1}}+\cdots +{{x}_{n}})}^{2}}$

is not hard to prove with quadratic root discriminant.

But the problem is that my computer science roommate want me to give an induction method. So I tried a new method to prove it, in which I used a small trick deserving taking a note.

Let me firstly restate the problem here: it’s known that

${{x}_{1}}+\cdots +{{x}_{n}}=1$, prove that ${{x}_{1}}^{2}+\cdots+{{x}_{n}}^{2}\geq \frac{1}{n}$

The equality holds if and only if

${{x}_{1}}={{x}_{2}}=\cdots ={{x}_{n}}$

when

$n=1,2$,

the inequality obviously holds true.

Assume that when

n=k,

the inequality holds true.

Then when

n=k+1,

if  k+1=2m,

${{x}_{1}}^{2}+\cdots +{{x}_{n}}^{2}=\sum_{i=1}^{2m}{{{x}_{i}}^{2}}=\sum_{i=1}^{m}{({{x}_{i}}^{2}+{{x}_{i+m}}^{2})}\geq \sum_{i=1}^{m}{({{(\frac{{{x}_{i}}+{{x}_{i+m}}}{2})}^{2}}+{{(\frac{{{x}_{i}}+{{x}_{i+m}}}{2})}^{2}})}\geq \frac{1}{2}\sum_{i=1}^{m}{{{({{x}_{i}}+{{x}_{i+m}})}^{2}}}\geq \frac{1}{2}m{{(\frac{\sum_{i=1}^{2m}{{{x}_{i}}}}{m})}^{2}}=\frac{1}{2m}=\frac{1}{n}$

if k+1=2m-1,

Therein lies the problem, we cannot break the formula into two parts equally if induction is used.

Instead of proving this inequality directly, we consider another problem which has a subtle link with the  issue we are dealing with. Say, when

${{x}_{1}}+\cdots +{{x}_{2m}}=1+\frac{1}{2m-1}$

Prove that

$\sum_{i=1}^{2m}{({{x}_{i}}^{2})}\geq 2m{{(\frac{1}{2m-1})}^{2}}$.

Actually, by induction method, because

m<2m-1

this statement holds true definitely. Further more, it gives

$\sum_{i=1}^{2m-1}{({{x}_{i}}^{2})}\geq \frac{1}{(2m-1)}$ when $\sum_{i=1}^{2m-1}{{{x}_{i}}}=1$.

Otherwise, the new problem value would be even smaller.

The inspring point for this method is to add a free variable to cater the old problem, though technical when applying it. And I hope this trick will bring me more stuff.