Dahn Jahn
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 Nov 1 asked Independence of Random Variables By Guessing Nov 1 revised What does $\bigcup\mathcal{P}A = A$ mean in English? Equality short enough for the title, I think it would make it more clear (and find it unlikely someone would find "this" useful) Nov 1 suggested approved edit on What does $\bigcup\mathcal{P}A = A$ mean in English? Nov 1 awarded Civic Duty Oct 30 comment Show that collection of finite dimensional cylinder sets is an algebra but not $\sigma$-algebra Ah! Heureka moment. Still need some clearing up, but that's mostly to be done in my mind and definitely outside a lengthy comment section, the main point's gotten through. Thanks! Oct 30 comment Show that collection of finite dimensional cylinder sets is an algebra but not $\sigma$-algebra Well, yes, but say $C_1 \cup C_2 = \{1,2\} \times \mathbb R^\mathbb N$ in my understanding. Oct 30 comment Show that collection of finite dimensional cylinder sets is an algebra but not $\sigma$-algebra But isn't that just $\mathbb N \times \mathbb R^\mathbb N$? This is what confuses me, $\{3\} \cup \mathbb R = \mathbb R$, right? And $\mathbb N$ is Borel, so what am I missing. Oct 30 comment product of random variables with different distributions What he's using there is $P(A|B)P(B) = P(A\cap B) = P(A) P(B)$ where the first equality is from the definition of conditional probability and the second from indepenence of $A$ and $B$ Oct 30 answered product of random variables with different distributions Oct 30 revised Prove $\sum^n_{i=1} (2i-1)=n^2$ by induction Added MathJax Oct 30 suggested approved edit on Prove $\sum^n_{i=1} (2i-1)=n^2$ by induction Oct 30 comment product of random variables with different distributions I am sure someone else will supply the math, but intuitively, how would you expect the distribution to behave? Such thinking may help greatly with formulating a more rigorous answer. Oct 30 comment Show that collection of finite dimensional cylinder sets is an algebra but not $\sigma$-algebra You say "it might be easier" - is it even possible to prove this through unions? Oct 29 asked When does continuity with probability one imply mean-square continuity Oct 27 accepted Why do we care about bijections in contability? Oct 27 comment Why do we care about bijections in contability? I suppose my confusion led me to pose the wrong question - perhaps what I really should've asked is why does the definition of countably infinite even consider a bijection, intuitively there seems to be no need for it, we only need to know whether $A$ is "smaller" (or better isn't "bigger") than $\mathbb N$. If you could comment on this a bit, that would be great, either way, I will accept this answer, as it answers my question. Oct 27 asked Why do we care about bijections in contability? Oct 25 comment Proving Independence Intuitively Thanks, I agree with the equality, but I am still not there, how does this prove that the distributions are independent? Oct 25 comment Proving Independence Intuitively I guess what I'd like to see is some mathematical notation, as the problem is precisely the perceived tenuous link between english words (no matter how convinced I am of their validity) and mathematical notation that I know how to work with and can identify as a sufficient proof Oct 25 comment Proving Independence Intuitively But why is it sufficient? Good points about $a.s$, thanks, it's good to never forget that, my brain just automatically disregards anything of measure zero.