“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 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.

Q5: How to construct a r.v. on one probability space that it has the desired distribution?

You can search for a THM called Skorokhod’s Representation THM, which is one of my most favourite THM in Probability. Read the proof and find sth amazing.

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)