In question 8a), M_{t} looks like the Dolean-Dades exponential martingale, which solves the basic differential equation of geometric Brownian motion, and its expectation is always one.
Very much turning into my favorite channel, even though I can I know what some of the notation means these questions seem like they're in a different language. xD
It's relieving that at least this exam seems to give some freebies (e.g. the ones about definitions), and the problems seem to be short or with hints on what theorem to apply. It's very unlike the real analysis exam which made you memorize the proofs of all different theorems and also test your problem solving abilities to the max.
The sequence where X_n = n in the interval [0,1/n) and 0 everywhere else doesn't converge to anything in L^1. The only thing it could converge to is the 0 function (after all, if there is a set of positive measure where the limit function f differs from 0 then at some point most of that set is outside the interval [0,1/n) and the L^1 norm of X_n - f would be strictly positive for all n after that). But the function does not converge to 0 either because it would need to converge to 0 in norm and the norm of X_n is 1 all the time.
Dominican Graduate Student here from an MSc in Applied Math, taking stochastic calculus now and definitely seeing how much I have left in probability haha
I'm told by Michio Kaku...there is a probability that your body could melt and reassemble on the other side of a brick wall, and gives his grad students that problem (he knows the answer) lol
This is more in my lane, as a masters degree in mathematical finance. I had to take the PhD level probability course twice (was an optional course for me, but glad I took it), to improve my grade from D to B. About 3000 thorems to remember and be able to prove.
No, there's no way 5 is that easy, right?. The X_n are given to be independent, and we are given the distributions of each which are identical. Hence let the common distribution be X. We have E(X) = 0, E(X^2) = 1/2 = Var(X). If S_n is defined as customary (the sum of the first n random variables), then by the (classical) central limit theorem, S_n/n -> N(E(X),Var(X)/n) in distribution. Hence, S_n/sqrt(n) -> sqrt(n)N(E(X),Var(X)/n) = N(E(X)sqrt(n),Var(X)) = N(0,1/2) in distribution
@@oroboros4858 I had a probability class in undergrad that covered everything in that PhD exam except for brownian motion and the exam in that undergrad course was harder (at least the harder than the questions that aren't about brownian motion since I can't judge the difficulty of that question). You don't need a PhD to think that this exam is very easy.
@@oroboros4858 I'm not from the US so I can't really comment on that. In my country people take more classes in undergrad compared to the US and I think someone that studied finance will need to do a lot of catchup. Idk how different it would be in the US. I think the struggling grad student said he was an enviromental science major in undergrad so it seems like going to grad school in a different topic is more common in the US compared to my country.
In question 8a), M_{t} looks like the Dolean-Dades exponential martingale, which solves the basic differential equation of geometric Brownian motion, and its expectation is always one.
Very much turning into my favorite channel, even though I can I know what some of the notation means these questions seem like they're in a different language. xD
It's relieving that at least this exam seems to give some freebies (e.g. the ones about definitions), and the problems seem to be short or with hints on what theorem to apply. It's very unlike the real analysis exam which made you memorize the proofs of all different theorems and also test your problem solving abilities to the max.
Man. I’m
So scared for this when I take this in my stats PhD
The sequence where X_n = n in the interval [0,1/n) and 0 everywhere else doesn't converge to anything in L^1. The only thing it could converge to is the 0 function (after all, if there is a set of positive measure where the limit function f differs from 0 then at some point most of that set is outside the interval [0,1/n) and the L^1 norm of X_n - f would be strictly positive for all n after that). But the function does not converge to 0 either because it would need to converge to 0 in norm and the norm of X_n is 1 all the time.
Dominican Graduate Student here from an MSc in Applied Math, taking stochastic calculus now and definitely seeing how much I have left in probability haha
please do a day in the life video
I'm told by Michio Kaku...there is a probability that your body could melt and reassemble on the other side of a brick wall, and gives his grad students that problem (he knows the answer) lol
Me just finishing calc 1 tryna understand this 😅
Me too Lmao 😂
This is more in my lane, as a masters degree in mathematical finance. I had to take the PhD level probability course twice (was an optional course for me, but glad I took it), to improve my grade from D to B. About 3000 thorems to remember and be able to prove.
3000?!?! I have a new found respect for you guys
@@Myrus_MBG Not literally 3000, but it felt like a lot
No, there's no way 5 is that easy, right?.
The X_n are given to be independent, and we are given the distributions of each which are identical. Hence let the common distribution be X. We have E(X) = 0, E(X^2) = 1/2 = Var(X).
If S_n is defined as customary (the sum of the first n random variables), then by the (classical) central limit theorem, S_n/n -> N(E(X),Var(X)/n) in distribution. Hence, S_n/sqrt(n) -> sqrt(n)N(E(X),Var(X)/n) = N(E(X)sqrt(n),Var(X)) = N(0,1/2) in distribution
Problem 3 is interesting
Hearing convergence outside of a real analysis context is frightening
WoW. The whole comment section is smarter than me
Nice
my brain hurt :(
problem 2 is so easy lol
What probability theory and statistics books would you recommend?
For probability, I would recommend Chung or Billingsley.
What university do you go to
I am finishing a bachelor degree in Statistics and in my opinion I could pass this test with 1 month of study. I thought it was way harder
👌🏼
this seems way too trivial for a PhD exam
this is almost far too easy for my skill level. actually, on second examination, this is definitely too easy for my skill level.
Do you have a PhD?
@@oroboros4858 I had a probability class in undergrad that covered everything in that PhD exam except for brownian motion and the exam in that undergrad course was harder (at least the harder than the questions that aren't about brownian motion since I can't judge the difficulty of that question). You don't need a PhD to think that this exam is very easy.
@@uriaviad9617 do you think it’s possible to do graduate studies in math if your undergraduate is finance?
@@oroboros4858 I'm not from the US so I can't really comment on that. In my country people take more classes in undergrad compared to the US and I think someone that studied finance will need to do a lot of catchup. Idk how different it would be in the US. I think the struggling grad student said he was an enviromental science major in undergrad so it seems like going to grad school in a different topic is more common in the US compared to my country.
A sophomore in my Statistics undergraduate course can answer these.
Who cares