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2h
revised Show that $2$ is not prime in $\mathbb Z[\sqrt{-d}]$ for odd prime $d$
added 11 characters in body
2h
answered Show that $2$ is not prime in $\mathbb Z[\sqrt{-d}]$ for odd prime $d$
2h
comment Determine the distribution
I divided $520$ by $52$ and got $20$.
2h
comment Length of pieces of stick broken at random
Hint: Let $Y$ be as in the comment. We want the mean of $|1-2Y|$, which is the same as the mean of $1-2Y$ given $Y\lt 1/2$.
4h
comment Determine the distribution
You are probably intended to say Poisson parameter $20$. Not a good model, there could be significant seasonal variation.
4h
comment Joint probabilty density function
You are welcome.
9h
comment Joint probabilty density function
We have $v=\frac{-1}{1+y}e^{-x(1+y)}$. I am not writing a full answer because that upsets some people on this site, and I do not like to hurt people's feelings.
9h
comment Joint probabilty density function
Integrate by parts, $u=x$, $dv=e^{-x(1+y)}\,dx$.
10h
comment Geometric proof of complex number equation
We want a point on the standard unit circle which is at distance $\sqrt{2}$ from $1$. So we want $x^2+y^2=1$, $(x-1)^2+y^2=2$. Subtract.
10h
comment Prove and disprove the following inequality.
Let $a=-3$, and $b=1$.
10h
comment Show that the solution for the Diophantine equation $x^2 - y^2 = N$ is unique if and only if $|N|$ or $\frac{|N|}{4}$, respectively, is $1$ or prime.
"Unique" needs to be defined, since even if we do not allow negatives, we can interchange $x$ and $y$ and get, most of the time, a "new" solution. But even if we do that, there are other cases of non-uniqueness. For example $x^2+y^2=45$ has a "unique" solution. Or more boringly $x^2+y^2=9$.
11h
comment Finding the probability density function of $U=Y_1+Y_2$
Finally, for $1\lt u\lt 2$, if we use an integral we have to break it up, first part $\int_0^{u-1}\int_0^1\,dy_2\,dy_1$, second part $\int_{u-1}^1\int_0^{u-y_1}\,dy_2\,dy_1$. But far clearer by geometry $F_U(u)$ is $1$ minus the area pf an isosceles right triangle with leg $1-u$, so we get $1-(1-u)^2$, density $2-u$. Sorry about the cramped exposition, comments have typesetting limitations.
11h
comment Finding the probability density function of $U=Y_1+Y_2$
Let $F_U(u)$ be the cdf of $U$. As you saw, $F_U(u)=0$ if $u\lt 0$, so there $f_U(u)-0$. If $u\gt 2$, it is clear that $F_U(u)=\Pr(U\le u)=1$ so $f_U(u)=0$. If $0\lt u\lt 1$, then $F_U(u)=\int_{y_1=0}^u \int_{y_2=0}^{u-y_1}\,dy_2\,dy_1$. (We are integrating over the part of the unit square that is "below" the line $y_1+y_2=u$.) More simply, we want the area of a certain isosceles right triangle with legs $u$. We get $\frac{u^2}{2}$ so density $u$. (More coming)
11h
comment Finding the probability density function of $U=Y_1+Y_2$
Wii try to comment later,to help you on your way. This question has been abswered on MSE several times, so I will not write a formal answer. Your expectation calculation is correct, or we could use symmetry.
13h
comment How important is the own talent for research of your PhD supervisor?
You are far more likely to be directed towards a doable problem. And when it comes to applying for a postdoc or a position, your supervisor's recommendation will pave the way. Also, a well-known person will likely have a research group that you can learn from.
13h
reviewed Leave Open Logical bulk pricing rate decrease
13h
reviewed Leave Open Defining addition in second order logic
13h
comment How important is the own talent for research of your PhD supervisor?
Absolutely critical.
13h
comment Could someone take a crack at this number theory problem?
Let the elements of $X$ be $x_1,x_2,\dots,x_m$. Since this is a complete residue system, if $1\le i\lt j\le m$, then $x_i\not\equiv x_j$ modulo $m$. We need to show that all the $ax_i$ are incongruent modulo $m$. Suppose to the contrary that there exist $1\le i\lt j\le m$ such that $ax_i\equiv ax_j\pmod{m}$. Then since $a$ and $m$ are relatively prime, we have $x_i\equiv x_j\pmod{m}$, which is false.
13h
comment Finding the probability density function of $U=Y_1+Y_2$
For expectation we really don't need (1), since by the linearity of expectation $E(Y_1+Y_2)=E(Y_1)+E(Y_2)$. For the density your case splitting is good. Are you going to use a convolution to find the density directly, or will you go through the cumulative distribution function?