User StubbornAtom

36 votes

6 answers

6k views

What is the most rigorous definition of a matrix?

asked Jun 4, 2016 at 5:33

14 votes

1 answer

432 views

Evaluating $\int _{-100}^{100}\lfloor {x^3}\rfloor \,dx$

asked Jul 2, 2016 at 12:05

13 votes

4 answers

561 views

Distribution and moments of $\frac{X_iX_j}{\sum_{i=1}^n X_i^2}$ when $X_i$'s are i.i.d $N(0,\sigma^2)$

asked Nov 15, 2021 at 14:21

12 votes

1 answer

3k views

Definiteness of a general partitioned matrix $\mathbf M=\left[\begin{matrix}\bf A & \bf B\\\bf B^\top & \bf D \\\end{matrix}\right]$

asked May 14, 2017 at 15:37

12 votes

1 answer

3k views

Showing $E(S^2\mid \bar X)=\bar X$ for i.i.d Poisson random variables $X_i$

asked Aug 13, 2018 at 20:24

10 votes

3 answers

3k views

Number of integer triplets $(a,b,c)$ such that $a<b<c$ and $a+b+c=n$

asked Jul 22, 2016 at 11:28

10 votes

4 answers

7k views

Maximum area of a rectangle inscribed in a triangle is $1/2$ the area of triangle

asked Aug 17, 2016 at 16:23

7 votes

4 answers

554 views

How to find $\max\int_{a}^{b}\left (\frac{3}{4}-x-x^2 \right )\,dx$ over all possible values of $a$ and $b$, $(a<b)$?

asked Jun 4, 2016 at 6:20

6 votes

2 answers

247 views

How to find $\sum_{A \subset S} (\min A)$ and $\sum_{A \subset S} (\max A)$ if $S=\{1,2,...,n\}$?

asked Jun 30, 2016 at 16:40

6 votes

3 answers

8k views

What is the logic behind the digit sum formula?

asked Nov 29, 2017 at 11:11

5 votes

2 answers

178 views

Rank/determinant of the $n\times n$ matrix $((a_{ij}))$ where $a_{ij}=(i+j-1)^2$

asked Feb 4, 2019 at 16:37

5 votes

1 answer

601 views

What is the distribution of $N-X_N$ if $X_i$'s are i.i.d $\operatorname{Exp}(1)$ and $N=\min\{n\ge1:X_n>1\}$?

asked Mar 12, 2020 at 19:16

5 votes

2 answers

1k views

If $A\subset\{1,11,21,31,...,541,551\}$ such that $\forall x,y\in A: x+y\ne 552$, then $|A|\le28$

asked Oct 31, 2016 at 8:12

5 votes

1 answer

88 views

Finding $\min f(x)$ where $f(x)=\int_0^1 |t-x|t\,dt \quad \forall x \in \mathbb{R}$ [duplicate]

asked Jul 14, 2016 at 17:36

4 votes

5 answers

1k views

How to evaluate $\int_0^{2\pi}\frac{1}{(1+a\cos {\theta})^2}\,d\theta$ without contour integration?

asked Jul 2, 2016 at 12:43

4 votes

3 answers

861 views

Unique pair of positive integers $(p,n)$ satisfying $p^3-p=n^7-n^3$ where $p$ is prime

asked Nov 12, 2016 at 14:01

4 votes

3 answers

244 views

Find $\lim_{n\to\infty}\frac{g(t+n)}n$ for $g(t)=\int_0^tf(x)\,dx$, where $f(x+1)=f(x)$

asked Jul 8, 2016 at 12:13

4 votes

3 answers

216 views

Simplifying $\big(\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}\big)/{\binom{M}{n}}$

asked Aug 11, 2016 at 18:38

4 votes

1 answer

457 views

Proving that if $f(x)=x$ has no real solution then $f(f(x))=x$ has no real solution either

asked Jun 4, 2016 at 11:14

4 votes

3 answers

150 views

How do I find $\lim_{n\to \infty}\left [\left (1+\frac{2}{n^a}\right )^{-n^b}n^c\right ]$ for real $a,b,c$ and $n\geq 1$?

asked Jun 16, 2016 at 7:17

4 votes

5 answers

993 views

Is saying 'limit does not exist' the same as saying 'limiting value is $\pm \infty $'?

asked Jun 16, 2016 at 8:25

4 votes

5 answers

1k views

All roots $\lambda$ of $\det(A-\lambda B)=0$ are $\ge1$ when $B$ is p.d and $A-B$ is n.n.d.

asked Feb 28, 2018 at 20:19

4 votes

1 answer

1k views

Is there a quick way to find the remainder when this determinant is divided by $5$?

asked Sep 18, 2016 at 11:56

4 votes

1 answer

62 views

Does $\frac1{n^\alpha}\sum_{k=1}^n kX_k$ converge in probability to a non-zero random variable?

asked Feb 13, 2022 at 8:44

3 votes

0 answers

92 views

What is $E\left[T_n(X_1,\ldots,X_n)\mid X_{(1)},\ldots,X_{(n)}\right]$ if $X_1,\ldots,X_n$ are i.i.d random variables?

asked Nov 6, 2021 at 20:31

3 votes

1 answer

82 views

Proving $(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$ for any events $A$ and $B$

asked Jan 26, 2017 at 7:43

3 votes

1 answer

154 views

For two continuous RVs $X$ and $Y$ such that $Y=X$, does $P(X=Y)=1$ hold?

asked Jul 3, 2017 at 18:54

3 votes

1 answer

118 views

Evaluating $\iint xy\,\mathrm{d}x\,\mathrm{d}y$ over an elliptical disk

asked Jan 21, 2018 at 8:32

3 votes

0 answers

219 views

Distribution of $ZX+(1-Z)Y$ where $X,Y\sim\mathcal N(0,1)$ and $Z\sim\mathcal U(0,1)$ are independent

asked Mar 21, 2018 at 11:40

3 votes

0 answers

129 views

Showing $(x^\top A^n x)(x^\top x)^{n-1}\geqslant(x^\top Ax)^n$ for any $n\in\mathbb N$ where $A$ is a symmetric matrix

asked May 15, 2018 at 10:20

Stack Exchange Network

78 Questions

What is the most rigorous definition of a matrix?

Evaluating $\int _{-100}^{100}\lfloor {x^3}\rfloor \,dx$

Distribution and moments of $\frac{X_iX_j}{\sum_{i=1}^n X_i^2}$ when $X_i$'s are i.i.d $N(0,\sigma^2)$

Definiteness of a general partitioned matrix $\mathbf M=\left[\begin{matrix}\bf A & \bf B\\\bf B^\top & \bf D \\\end{matrix}\right]$

Showing $E(S^2\mid \bar X)=\bar X$ for i.i.d Poisson random variables $X_i$

Number of integer triplets $(a,b,c)$ such that $a<b<c$ and $a+b+c=n$

Maximum area of a rectangle inscribed in a triangle is $1/2$ the area of triangle

How to find $\max\int_{a}^{b}\left (\frac{3}{4}-x-x^2 \right )\,dx$ over all possible values of $a$ and $b$, $(a<b)$?

How to find $\sum_{A \subset S} (\min A)$ and $\sum_{A \subset S} (\max A)$ if $S=\{1,2,...,n\}$?

What is the logic behind the digit sum formula?

Rank/determinant of the $n\times n$ matrix $((a_{ij}))$ where $a_{ij}=(i+j-1)^2$

What is the distribution of $N-X_N$ if $X_i$'s are i.i.d $\operatorname{Exp}(1)$ and $N=\min\{n\ge1:X_n>1\}$?

If $A\subset\{1,11,21,31,...,541,551\}$ such that $\forall x,y\in A: x+y\ne 552$, then $|A|\le28$

Finding $\min f(x)$ where $f(x)=\int_0^1 |t-x|t\,dt \quad \forall x \in \mathbb{R}$ [duplicate]

How to evaluate $\int_0^{2\pi}\frac{1}{(1+a\cos {\theta})^2}\,d\theta$ without contour integration?

Unique pair of positive integers $(p,n)$ satisfying $p^3-p=n^7-n^3$ where $p$ is prime

Find $\lim_{n\to\infty}\frac{g(t+n)}n$ for $g(t)=\int_0^tf(x)\,dx$, where $f(x+1)=f(x)$

Simplifying $\big(\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}\big)/{\binom{M}{n}}$

Proving that if $f(x)=x$ has no real solution then $f(f(x))=x$ has no real solution either

How do I find $\lim_{n\to \infty}\left [\left (1+\frac{2}{n^a}\right )^{-n^b}n^c\right ]$ for real $a,b,c$ and $n\geq 1$?

Is saying 'limit does not exist' the same as saying 'limiting value is $\pm \infty $'?

All roots $\lambda$ of $\det(A-\lambda B)=0$ are $\ge1$ when $B$ is p.d and $A-B$ is n.n.d.

Is there a quick way to find the remainder when this determinant is divided by $5$?

Does $\frac1{n^\alpha}\sum_{k=1}^n kX_k$ converge in probability to a non-zero random variable?

What is $E\left[T_n(X_1,\ldots,X_n)\mid X_{(1)},\ldots,X_{(n)}\right]$ if $X_1,\ldots,X_n$ are i.i.d random variables?

Proving $(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$ for any events $A$ and $B$

For two continuous RVs $X$ and $Y$ such that $Y=X$, does $P(X=Y)=1$ hold?

Evaluating $\iint xy\,\mathrm{d}x\,\mathrm{d}y$ over an elliptical disk

Distribution of $ZX+(1-Z)Y$ where $X,Y\sim\mathcal N(0,1)$ and $Z\sim\mathcal U(0,1)$ are independent

Showing $(x^\top A^n x)(x^\top x)^{n-1}\geqslant(x^\top Ax)^n$ for any $n\in\mathbb N$ where $A$ is a symmetric matrix