Most Popular
1500 questions
65
votes
3 answers
product distribution of two uniform distribution, what about 3 or more
Say $X_1, X_2, \ldots, X_n$ are independent and identically distributed uniform random variables on the interval $(0,1)$.
What is the product distribution of two of such random variables, e.g.,
$Z_2 = X_1 \cdot X_2$?
What if there are 3; $Z_3 = X_1…
lulu
- 1,008
65
votes
9 answers
Do all square matrices have eigenvectors?
I came across a video lecture in which the professor stated that there may or may not be any eigenvectors for a given linear transformation.
But I had previously thought every square matrix has eigenvectors.
65
votes
2 answers
Does $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ converge?
Does $\sum _{n=1}^{\infty } \dfrac{\sin(\text{ln}(n))}{n}$ converge?
My hypothesis is that it doesn't , but I don't know how to prove it. $ζ(1+i)$ does not converge but it doesn't solve problem here.
Darius
- 1,334
65
votes
9 answers
Refuting the Anti-Cantor Cranks
I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real numbers, arguably one of the most beautiful ideas…
Keshav Srinivasan
- 10,204
65
votes
1 answer
Formal definition of conditional probability
It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space
$ (\Omega, \mathscr{A}, \mu ) $ with $\mu(\Omega) = 1 $, and a random variable $ X :…
smiley06
- 4,157
65
votes
10 answers
Is it not effective to learn math top-down?
By top-down I mean finding a paper that interests you which is obviously way over your head, then at a snail's pace, looking up definitions and learning just what you need and occasionally proving basic results. Eventually you'll get there but is…
Daniel Donnelly
- 21,342
65
votes
7 answers
How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?
I would like to investigate the convergence of
$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$
Or more precisely, let $$\begin{align}
a_1 & = \sqrt 1\\
a_2 & = \sqrt{1+\sqrt2}\\
a_3 & = \sqrt{1+\sqrt{2+\sqrt 3}}\\
a_4 & =…
MJD
- 65,394
- 39
- 298
- 580
65
votes
6 answers
What is the general equation of the ellipse that is not in the origin and rotated by an angle?
I have the equation not in the center, i.e.
$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$
But what will be the equation once it is rotated?
andikat dennis
- 929
65
votes
2 answers
Wild automorphisms of the complex numbers
I read about so called "wild" automorphisms of the field of complex numbers (i.e. not the identity nor the complex conjugation). I suppose they must be rather weird and I wonder whether someone could explain in the simplest possible way (please) how…
Gerard
- 1,513
65
votes
7 answers
Is there any geometric intuition for the factorials in Taylor expansions?
Given a smooth real function $f$, we can approximate it as a sum of polynomials as
$$f(x+h)=f(x)+h f'(x) + \frac{h^2}{2!} f''(x)+ \dotsb = \sum_{k=0}^n \frac{h^k}{k!} f^{(k)}(x) + h^n R_n(h),$$
where $\lim_{h\to0} R_n(h)=0$.
There are multiple ways…
glS
- 6,818
65
votes
5 answers
Is the box topology good for anything?
In point-set topology, one always learns about the box topology: the topology on an infinite product $X = \prod_{i \in I} X_i$ generated by sets of the form $U = \prod_{i \in I} U_i$, where $U_i \subset X_i$ is open. This seems naively like a…
Nate Eldredge
- 97,710
65
votes
1 answer
Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$
Prove without evaluating the integrals that:$$2\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx=\int_\frac{\pi}{2}^\pi\frac{x\ln(1-\sin x)}{\sin x}dx\label{*}\tag{*}$$
Or equivalently:
$$\boxed{\int_0^\pi\frac{x\ln(1-\sin x)}{\sin…
Zacky
- 27,674
65
votes
2 answers
What is the difference between minimum and infimum?
What is the difference between minimum and infimum?
I have a great confusion about this.
Manoj
- 1,757
65
votes
7 answers
Compute polynomial $p(x)$ if $x^5=1,\, x\neq 1$ [reducing mod $\textit{simpler}$ multiples]
The following question was asked on a high school test, where the students were given a few minutes per question, at most:
Given that,
$$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$
and,
$$Q(x)=x^4+x^3+x^2+x+1$$
what is the remainder of $P(x)$…
joeblack
- 1,013
65
votes
6 answers
Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
This is being asked in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions, and here: List of abstract duplicates.
What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty}…
Eric Naslund
- 72,099