2
votes
2answers
123 views

Bounds for the fat tail after trimming the mean?

I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$. 1) Does this quantity $f(X,t)$ have a name? ...
4
votes
0answers
139 views

Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all $n\geq3$, the function: ...
3
votes
0answers
173 views

Maximization of a total variation distance subject to another total variation distance in Markov chain

Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
5
votes
1answer
106 views

Deviation bound for the maximum of the norm of Wiener process

Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof: $$ ...
1
vote
0answers
119 views

Placing Bounds on Correlation/Covariance Through Correlation with an Intermediate Variable

I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, but I think it is ...
0
votes
1answer
2k views

Why does sample standard deviation underestimate population standard deviation? [closed]

I am aware of Jensen's inequality where, given the concave square root function, the mean of the square root is lesser than the square root of the mean. However, I cannot figure out why the square ...
0
votes
1answer
311 views

Minimum distance between two data sets

Suppose we have two sets of data, $X$ and $Y$, each of which contains $10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$, $x_{1}\ge\cdots\ge x_{10}>0$ and ...
0
votes
0answers
177 views

Two-sample t-test variant?

I have a situation that is the opposite of the usual two-sample t-test for equal means: I'm trying to show that it's statistically unlikely that two samples are different. Is there a statistical test ...
1
vote
1answer
743 views

Probability inequalities

Hi everyone, I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an ...
1
vote
1answer
328 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 ...
1
vote
1answer
164 views

Maximal inequality over two indices

In Freedman's series of 3 books on Markov processes, I find that I keep on running into terms like: P[$\max_{0 \leq s \leq 1, s \leq t \leq rs}$ | B(t) - B(s) | > $\epsilon$] in the background of ...
5
votes
2answers
991 views

Inequality involving probability measures [closed]

I have been working on a problem(alternate minimization) where I want to establish an inequality in which I am stuck. An $\alpha$- parameterized version of the divergence(Kullback-Leibler) takes the ...