Intereting Posts

Inverse of $y=xe^x$
The number of Sylow subgroups on $G$ with $|G|=pqr$
Evaluation of $\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\,\mathrm{d}x$
This one weird thing that bugs me about summation and the like
Express roots in polynomials of equation $x^3+x^2-2x-1=0$
Finite abelian groups – direct sum of cyclic subgroup
Proof that $a\mid b \land b\mid c \Rightarrow a\mid c $
Proofs involving $ (A\setminus B) \cup (A \cap B)$
This sentence is false
Impossibility of theories proving consistency of each other?
Generalized rotation matrix in N dimensional space around N-2 unit vector
Is the relation on integers, defined by $(a,b)\in R\iff a=5q+b$, a function?
Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$?
A counter example of best approximation
Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$?

A square with side $a$ is given. What is the average distance between two uniformly-distributed random points inside the square?

For more general rectangle case, see here. The proof found there is fairly complex, and I looking for simpler proof for this special case

See also line case

- Geometric mean of prime gaps?
- What does the $L^p$ norm tend to as $p\to 0$?
- Prove inequality of generalized means
- Average bus waiting time
- Very fascinating probability game about maximising greed?
- Correctness of a statistical evaluation of a parameter

- Is it possible to imitate a sphere with 1000 congruent polygons?
- Finding a point along a line a certain distance away from another point!
- What is the most elegant proof of the Pythagorean theorem?
- Project Euler Question 222
- How many circles are needed to cover a rectangle?
- Dividing $n$ gon into 4 equal parts
- How to find intersection of two lines in 3D?
- Radius of circumscribed circle of triangle as function of the sides
- Equal perimeter and area
- Smallest possible triangle to contain a square

We just have to compute:

$$ I=\int_{[0,1]^4}\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\,d\mu. \tag{1}$$

Assuming that $X_1$ and $X_2$ are two independent random variables, uniformly distributed over $[0,1]$, the pdf of their difference $\Delta X=X_1-X_2$ is given by:

$$ f_{\Delta X}(x) = \left(1-|x|\right)\cdot\mathbb{1}_{[-1,1]}(x)\tag{2}$$

hence:

$$\begin{eqnarray*} I &=& \iint_{[-1,1]^2}(1-|x|)(1-|y|)\sqrt{x^2+y^2}\,dx\,dy \\&=&4\iint_{[0,1]^2}xy\sqrt{(1-x)^2+(1-y)^2}\,dx\,dy\tag{3}\end{eqnarray*}$$

that is tedious to compute but still possible; we have:

$$ I = \frac{2+\sqrt{2}+5\operatorname{arcsinh}(1)}{15}=\frac{2+\sqrt{2}+5\log(1+\sqrt{2})}{15}=0.52140543316472\ldots$$

(OEIS A091505)

hence the average distance between two random points in $[0,a]^2$ is around the $52.14\%$ of $a$.

- Integral:$ \int^\infty_{-\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}dx $
- Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?
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- Why can't the Polynomial Ring be a Field?
- Is there a (deep) relationship between these various applications of the exponential function?
- Is there a subfield $F$ of $\Bbb R$ such that there is an embedding $F(x) \hookrightarrow F$?
- Convergence of $np(n)$ where $p(n)=\sum_{j=\lceil n/2\rceil}^{n-1} {p(j)\over j}$
- Prove that the degree of the splitting field of $x^p-1$ is $p-1$ if $p$ is prime
- The structure of a finite group of order 40320
- Understanding of extension fields with Kronecker's thorem
- Examples of a monad in a monoid (i.e. category with one object)?
- Integral $\int_{-\infty}^{\infty}\frac{\cos(s \arctan(ax))}{(1+x^2)(1+a^2x^2)^{s/2}}dx$
- Interchanging the order of differentiation and summation
- Exponential Function as an Infinite Product
- Is $\mathbb R$ terminal among Archimedean fields?