@TedShifrin I knew these were important, but we brushed over it too quickly. I didn't have time to soak it in. Now I see how one establishes this stuff for the non 2D case. Wow, and it relates to calculus!
@Chris'ssistheartist seems the version with $H_n$ comes with $\psi^{(1)}\left(\frac{1}{3}\right),\psi^{(1)}\left(\frac{2}{3}\right)$ and $H_n^{(2)}$ comes with $\psi^{(2)}\left(\frac{1}{3}\right),\psi^{(2)}\left(\frac{1}{3}\right)$ .. Interesting indeed :D
Is it true that for two continuous random variables, $$P(Y\leqslant t)=\int_{\Bbb R}P(Y\leqslant t\mid X=s)f_X(s)ds$$ where $f_X$ is the density function of $X$?
3500 for now, @Cbjork, just to fill any and all gaps. I've noticed some gaping holes in my knowledge, or at least my base of intuition, by working through differential geometry.
@Fargle it's a fun class. My favorite problem from that class is prove or give a counterexample: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R} $ a smooth function have one critical point $\vec{c}$ where $f(\vec{c})$ is a local min. $f(\vec{c})$ is a global min.
@PedroTamaroff Wouldn't it just be the conditional probability density function for $Y$ given $X=x$ times the probability density function for $X$? That would be $\frac{1}{60-x} \cdot \frac{1}{20}$.
@PedroTamaroff $$\frac1{20}\int_0^{20}\frac1{60-x}\int_x^{60}f(x,y)\,\mathrm{d}y\,\mathrm{d}x$$ is the mean of $f(x,y)$ given the $X$ and $Y$ you described: $X$ uniform on $[0,20]$ and $Y$ uniform on $[x,60]$
@PedroTamaroff $$P(a\le Y\le b|X=x)=\begin{cases}0 & a\le b\le x \\ \frac{b-x}{60-x} & a\le x\le b \\ \frac{b-a}{60-x} & x\le a\le b \end{cases}$$ Divide by $b-a$ and then let $b\to a^+$. If $a< x$ this will be $0$, and if $x\le a$ it will be $\frac{1}{60-x}$. Now multiply by the probability density associated with $X=x$ (namely $\frac{1}{20}$).
I understand that some solutions to the classical gravitational 3-body problem have been found to be periodic, such as the choreograph below:
Could it be possible that all solutions to the 3-body problem turn out to be periodic given enough time, or has it been shown that for some solutions, t...
@r9m I know how to do it now, referring to your series!!! :-) Well, I only need to write things down. And there is one more thing: it seems one need to also know to calculate
@BalarkaSen The nice thing about nets is that most statements in metric spaces that can be formulated in terms of sequences become "correct" in general topological spaces when formulated with nets
such as Hausdorff iff nets converge to at most one point
@SohamChowdhury No, they are just a generalization of sequences
but they are often mentioned in connection with filters, which have a set-theoretic aspect due to the fact they there might not be any non-principal ultra-filters unless one assumes AC
I understand that some solutions to the classical gravitational 3-body problem have been found to be periodic, such as the choreograph below:
Could it be possible that all solutions to the 3-body problem turn out to be periodic given enough time, or has it been shown that for some solutions, t...
