“要多想。” ― 《三体》

Interesting Quizzes

Q1: For two sequence of r.v.’s (Xn) and (Yn) independent in a same probability space. If Xn→X,Yn→Y both in distribution. Do you think Xn+Yn→X+Y in distribution?

Think it carefully. You can modify the statement and make it more tricky.

Q2: How to prove any two families of o.n.b of one Hilbert space have the same cardinality?

You will find sth amazing in definition of “SUM”.

Q3: Do you really think p(θ∈(a,b))∈{0,1} for some parameter θ?

This is an essential difference between a Bayesian and a Frequenist. I have to say: I am totally a Bayesian.

Q4: For a given r.v. X, could you find a r.v. Y st. it’s i.i.d. as X?

Generally, we require the probability space to be nonatomic to guaratee the indepedence will hold. And You can search for a THM called Skorokhod’s Representation THM, which is one of my most favourite THM in Probability.

Q5: For a n.v.s E. F is a close subspace, is there a m in F st. d(x,F)=d(x,m)?

There’s a sufficient and necessary condition for its holding.

Q6: Do you think one transition function can correspond one CTMC? What’s about one generator and one CTMC?

It first amazes me because it’s a simple example that satisfies strong Markov property but not a Feller process. (Actually, Feller property is a stronger property than merely strong Markov) If you are interested in this question, you can search for a process called Blackwell process.

Q7: What’s the dual space of R^∞ equipped with box topology?

Can you recall the definition of dual space? You can think it algebracally rather than analysisly.(hint: search for concepts: inductive limit and projective limit)

Mein Glück

–Friedrich Nietzsche

“Seit ich des Suchens müde ward,

“自从厌倦于追寻，

Erlernte ich das Finden.

我已学会一觅即中。

Seit mir ein Wind hielt Widerpart,

自从一股逆风袭来，

Segl’ ich mit allen Winden.”

我已能抵御八面来风架舟而行。”