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location Montreal
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visits member for 3 years, 11 months
seen Aug 20 at 13:43

Computer scientist student who is trying to study math and trying avoid asking stupid questions on this website.


May
13
comment If the sum of two independent random variables is in $L^2$, is it true that both of them are in $L^1$?
@Batman That's why proving $X$ and $Y$ are in $L^1$ should be an easier task, isn't it?
May
13
comment If the sum of two independent random variables is in $L^2$, is it true that both of them are in $L^1$?
Can you give an example s.t. $\mathbb E(X+Y)^2 < \infty$ but $\mathbb E X = \infty$ and $\mathbb E Y = \infty$?
May
12
comment 1D biased random walk - is the event of infinte many returns a tail event?
You should check Hewitt–Savage zero–one law - en.wikipedia.org/wiki/…
May
7
comment hat problem and probability
If the colors are assigned at equal probability and independently, you will not be able do better than 1/2.
Nov
17
comment The limit of integer valued random variables must be integer valued?
OK. BTW, does my example shows that the portmanteau theorem works only for random variables on $\mathbb R$?
Nov
17
comment The limit of integer valued random variables must be integer valued?
It's not clear whether the author meant to define $D_n$ and $D$ in $\mathbb R$ or compactification of $\mathbb R$. That's a question that comes to me again and again -- When the author says "random variables", does he/she allow them to take $\infty$ as value or not?
Nov
17
comment The limit of integer valued random variables must be integer valued?
Thanks. I'm actually aware of this theorem and it is how I thought I had proved it. But when I came up with the "counter example" that I got confused again. Can we say that $\infty \in \mathbb Z$?
Aug
31
comment Martingale with bounded increment.
@Did, I will remember to do that next time!
Jun
19
comment Is it possible to have $\mathbb E S_n \to \infty$ but $\inf S_n = -\infty$ a.s.?
@ByronSchmuland Hmm, you're right. $S_n / n \to 2p - 1 > 0$ almost surely, so $S_n \to \infty$ almost surely
Jun
19
comment Is it possible to have $\mathbb E S_n \to \infty$ but $\inf S_n = -\infty$ a.s.?
@ByronSchmuland Yeah, you're right about case 1. It should be $S_n = 0$ for all $n$.
Jun
17
comment How many ways can $5$ rings be placed on $4$ fingers?
Shouldn't this goes to infinity?
Jun
16
comment combinatorics - Distribution of Distinct Balls into Distinct Boxes
By nPk do you mean $\binom n k$?
Jun
16
comment Convergence to exponential function.
I see my mistake!
Jun
3
comment Scheffe’s Theorem
@ChrisJanjigian I see, it is $1 \le n \le \infty$. Thanks!
Jun
3
comment Scheffe’s Theorem
@ChrisJanjigian How do we get $\int f_\infty = 1$? Dominated convergence theorem?
May
31
comment Difference between $P(Y \ge y)$ and $P(Y > y)$.
Thanks! Somehow I believed the sum should starts from $0$ without much thinking.
May
30
comment Difference between $P(Y \ge y)$ and $P(Y > y)$.
As for the sum, I was thinking if $>$ and $\ge$ versions of Lemma 2.2.8 is equivalent. Then we should have $\mathbb E(Y) = \int_0^\infty P (Y \ge y) \, \mathrm dy = \sum_{y \ge 0} P(Y \ge y)$, for $Y$ takes only integer values which seems to be wrong.
May
30
comment Difference between $P(Y \ge y)$ and $P(Y > y)$.
So actually we can replace $>$ with $\ge$ in the proof? I'm confused because latter in the book, exercise 2.2.7, a generalized version of Lemma 2.2.8 is given, with $>$ replaced by $\ge$, that's why I'm considering whether there are difference between the two.
May
30
comment Difference between $P(Y \ge y)$ and $P(Y > y)$.
But the integral $\int_0^\infty py^{p-1} 1_{(Y > y)} \, \mathrm dy$ should not change if we replace $>$ with $\ge$, right? This is an integral with respect to Lebesgue measure, which shouldn't change when omitting a single point.
May
30
comment Difference between $P(Y \ge y)$ and $P(Y > y)$.
As you pointed out in the link, the sum should start from $y \ge 1$, not $y \ge 0$.