All Questions
12 questions
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
5
votes
2
answers
202
views
Monotonicity of a parametric integral
For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<...
- 128k
5
votes
2
answers
243
views
Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?
This has received no full solution at StackExchange.
As per https://dlmf.nist.gov/8.10#E13 we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\...
- 804
5
votes
1
answer
445
views
A two-point inequality
Let $M(p,q) = (2p-\sqrt{p^{2}+q^{2}})\sqrt{p+\sqrt{p^{2}+q^{2}}}$ and set $B(t) = M(x+t, \sqrt{t^{2}+(y+bt)^{2}})$. Given any real $x,y,b$ is it true that $\varphi(t) = B(t)+B(-t)$ is decreasing in $...
- 3,946
2
votes
1
answer
208
views
Proving an exponential sum inequality for symmetric Hamming distance sequences in binary vectors
Background: Let $X = \{0,1\}^k$ represent the set of all binary vectors of length $k$. For two binary vectors $x, y \in X$, the Hamming distance $d_H(x, y)$ is defined as the number of positions where ...
- 349
2
votes
1
answer
198
views
Bounds for the beta CDF
This question is closely related to a previous question that I asked here:
An inequality involving the beta distribution
Let $a,b$ be strictly positive integers, and let $F_{a,b}(x)$ denote the CDF ...
- 4,049
2
votes
1
answer
286
views
An inequality for abstract Cauchy problem
Consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x_0$, where $A$ generates an analytic $C_0$-semigroup on a Banach space $X$. How we can prove an inequality of ...
- 97
2
votes
1
answer
675
views
Moment generating function of random unit vector
Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of
$$\mathbb{E}[\exp(X^Tv)]$$
for any $v$?
- 1,495
2
votes
1
answer
248
views
Ratio of expectation involving random unit vectors
Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sum_{i=1}^nu_i^2v_i^2$, ...
- 1,495
1
vote
1
answer
148
views
An inequality about binomial distribution
Statement
Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that
$$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$
...
- 35
1
vote
1
answer
356
views
Statistical inequality
Let $X$ be a finite discrete variable and $X\ge0$. Is it true that
$$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$
where $\operatorname{...
- 95
1
vote
0
answers
107
views
$L^p$ inequality for "positively correlated" random variables
Suppose that we have $m$ complex-valued random variables $\xi_1,\ldots,\xi_m$ and assume the following "positive correlation" property: for all non-negative integers $\alpha_1,\ldots,\...
- 7,193