Skip to main content

All Questions

Filter by
Sorted by
Tagged with
2 votes
2 answers
411 views

Given a total variation distance from uniform, how well can we bound the probabilities of sub-intervals?

I made the following claim, which I now see that I don't know how to prove. Can anyone prove it? Claim. Let $f$ be a concave and non-negative function on $[0,1]$ with $$\int_0^1 f = 1,$$ $$\int_0^1 |f-...
user avatar
7 votes
2 answers
281 views

An inequality involving the beta distribution

Let $F$ be the CDF for a Beta distribution, $$F(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int_{0}^{x}t^{a-1}(1-t)^{b-1}\,dt$$ with $a,b\geq 1$. Is it true that $$\frac{b}{a+b} \leq\int_0^1\sqrt{F(x)}...
Tom Solberg's user avatar
  • 4,049
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 ...
Tom Solberg's user avatar
  • 4,049
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 $v$?
neverevernever's user avatar
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 $X=\sum_{i=1}^nu_i^2v_i^2$, ...
neverevernever's user avatar