User Fei Cao

4 votes

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Prove that $f(x) = \frac{5x^{2}}{1 + x^{2}}$ is bounded

answered Aug 20, 2020 at 22:15

4 votes

Prove that $0\le\int_0^1\log(u){\rm d}x+\frac1{2\pi^2}\int_0^1\frac1{u^2}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2{\rm d}x$

answered Mar 5, 2021 at 1:03

4 votes

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Covariance of Brownian motion intuition

answered Mar 7, 2021 at 1:28

4 votes

$V[\sqrt{X}]\le\frac{V[X]}{E[X]}$ for non-negative $X$

answered Jan 6, 2022 at 16:58

3 votes

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How do I show that $\left|x+4\right|<x$ is impossible?

answered Apr 23, 2021 at 18:33

3 votes

Why use exponentials in the Markov inequality?

answered May 28, 2021 at 15:27

3 votes

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Discrete stochastic process to stochastic differential equation

answered Jul 28, 2021 at 17:57

3 votes

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Show that $E_f(\log f) \ge E_f (\log g)$

answered Apr 10, 2021 at 23:27

3 votes

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Approximating Lipschitz Functions by Sigmoidal Functions

answered Apr 19, 2021 at 0:33

3 votes

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Why does $\sum_{n=1}^{∞}e^{-\frac{an}{x}}=\frac{1}{e^{\frac{a}{x}}-1}$?

answered Mar 7, 2021 at 3:34

2 votes

Infinitesimal generator of the Brownian motion on a sphere

answered Jul 26, 2021 at 23:19

2 votes

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prove a challenging inequality or find a counterexample to it

answered May 22, 2021 at 17:31

2 votes

$L_2$ distance and Hellinger distance

answered Oct 7, 2020 at 22:26

2 votes

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Lower bound for Strongly convex and Lipschitz gradient function

answered Feb 16, 2021 at 23:17

2 votes

Show that $f(x)=x+\frac{1}{x^2+1}$ is strictly increasing.

answered Dec 16, 2020 at 18:54

2 votes

How do I solve $\lim_{n \to \infty} \frac{n!}{2^{n+1}}$

answered Jan 19, 2021 at 6:23

2 votes

When is $E(X-\mu)^2 \ne E(X^2)-\mu^2$?

answered Apr 26, 2021 at 4:51

2 votes

Prove that $E(X) = \int_{0}^{a} (1-F_X(x))\,dx$

answered Apr 26, 2021 at 17:25

2 votes

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Proof of Cauchy's functional equation

answered Apr 26, 2021 at 17:57

2 votes

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Unit Circle is a 1-manifold in $\mathbf{R}^2$ and a function is not a coordinate patch

answered Apr 3, 2021 at 23:43

2 votes

Introductory Text to Partial Differential Equations

answered Apr 29, 2021 at 4:00

2 votes

Problems solving the SDE $dX_t = aX_tdt +\sigma dB_t$. Why don't Ito's lemma work?

answered May 1, 2021 at 0:08

2 votes

Poincare inequality for Poisson random variables

answered Dec 25, 2021 at 23:22

1 vote

Show $2 + \log\left(\frac{a^2}{pa^2 + (1-p)b^2}\right) - \frac{2pa^2}{pa^2 + (1-p)b^2} - (1-p)\frac{1}{1-2p}\log\frac{1-p}{p} \leq 0$ for $p\in [0,1]$

answered Dec 31, 2021 at 4:46

1 vote

Given $f: \mathbf{R} \to \mathbf{R}$ is continuous and $\lim_{x\to -\infty}f(x) = \infty = \lim_{x\to \infty} f(x)$, show $f$ attains its minimum.

answered Nov 16, 2021 at 22:26

1 vote

Interpretation of adapted process?

answered Oct 17, 2022 at 2:21

1 vote

Using the definition of the the operator to prove that the generator for Bronwnian motion on $S^2$ is $\frac{1}{2}\Delta$

answered Jul 26, 2021 at 6:26

1 vote

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SDE of translated process $X(t) + a$

answered Apr 25, 2021 at 23:26

1 vote

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Lower bound on the $\Phi$-entropy of a Gaussian variable

answered Nov 8, 2021 at 2:12

1 vote

Area derivative under curve shortening flow of surfaces

answered Jun 19, 2021 at 22:22

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72 Answers

Prove that $f(x) = \frac{5x^{2}}{1 + x^{2}}$ is bounded

Prove that $0\le\int_0^1\log(u){\rm d}x+\frac1{2\pi^2}\int_0^1\frac1{u^2}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2{\rm d}x$

Covariance of Brownian motion intuition

$V[\sqrt{X}]\le\frac{V[X]}{E[X]}$ for non-negative $X$

How do I show that $\left|x+4\right|<x$ is impossible?

Why use exponentials in the Markov inequality?

Discrete stochastic process to stochastic differential equation

Show that $E_f(\log f) \ge E_f (\log g)$

Approximating Lipschitz Functions by Sigmoidal Functions

Why does $\sum_{n=1}^{∞}e^{-\frac{an}{x}}=\frac{1}{e^{\frac{a}{x}}-1}$?

Infinitesimal generator of the Brownian motion on a sphere

prove a challenging inequality or find a counterexample to it

$L_2$ distance and Hellinger distance

Lower bound for Strongly convex and Lipschitz gradient function

Show that $f(x)=x+\frac{1}{x^2+1}$ is strictly increasing.

How do I solve $\lim_{n \to \infty} \frac{n!}{2^{n+1}}$

When is $E(X-\mu)^2 \ne E(X^2)-\mu^2$?

Prove that $E(X) = \int_{0}^{a} (1-F_X(x))\,dx$

Proof of Cauchy's functional equation

Unit Circle is a 1-manifold in $\mathbf{R}^2$ and a function is not a coordinate patch

Introductory Text to Partial Differential Equations

Problems solving the SDE $dX_t = aX_tdt +\sigma dB_t$. Why don't Ito's lemma work?

Poincare inequality for Poisson random variables

Show $2 + \log\left(\frac{a^2}{pa^2 + (1-p)b^2}\right) - \frac{2pa^2}{pa^2 + (1-p)b^2} - (1-p)\frac{1}{1-2p}\log\frac{1-p}{p} \leq 0$ for $p\in [0,1]$

Given $f: \mathbf{R} \to \mathbf{R}$ is continuous and $\lim_{x\to -\infty}f(x) = \infty = \lim_{x\to \infty} f(x)$, show $f$ attains its minimum.

Interpretation of adapted process?

Using the definition of the the operator to prove that the generator for Bronwnian motion on $S^2$ is $\frac{1}{2}\Delta$

SDE of translated process $X(t) + a$

Lower bound on the $\Phi$-entropy of a Gaussian variable

Area derivative under curve shortening flow of surfaces