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Fei Cao's user avatar
Fei Cao's user avatar
Fei Cao's user avatar
Fei Cao
  • Member for 5 years, 1 month
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4 votes
Accepted

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

4 votes

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

4 votes
Accepted

Covariance of Brownian motion intuition

4 votes
Accepted

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

4 votes

Why use exponentials in the Markov inequality?

4 votes
Accepted

Discrete stochastic process to stochastic differential equation

4 votes

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

3 votes
Accepted

Approximating Lipschitz Functions by Sigmoidal Functions

3 votes
Accepted

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

3 votes

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

3 votes

Infinitesimal generator of the Brownian motion on a sphere

3 votes
Accepted

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

2 votes

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

2 votes

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

2 votes

$L_2$ distance and Hellinger distance

2 votes
Accepted

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

2 votes
Accepted

Lower bound for Strongly convex and Lipschitz gradient function

2 votes

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

2 votes
Accepted

prove a challenging inequality or find a counterexample to it

2 votes
Accepted

Proof of Cauchy's functional equation

2 votes

Poincare inequality for Poisson random variables

2 votes

Introductory Text to Partial Differential Equations

2 votes

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

1 vote

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

1 vote
Accepted

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

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.

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 $p\in [0,1]$

1 vote

Prove or Disprove: $\sum_{n\geq 0} (3p^2_n - p_n - 2p^3_n) \leq 0$ for any probability mass ${\bf p} = (p_0,p_1,\ldots)$

1 vote
Accepted

Good books to learn Complex Analysis and Contour Integration?

1 vote

Interpretation of adapted process?