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 , assume that there exists a r.v , then we can measure the events by using . What’s more, adding a time dimension to make it a stochastic process , then we have properties like 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……
In mathematics, the inequality
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
, prove that
The equality holds if and only if
the inequality obviously holds true.
Assume that when
the inequality holds true.
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
Actually, by induction method, because
this statement holds true definitely. Further more, it gives
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.