3
votes
0answers
116 views

Total variation and Hellinger distance inequality between truncated Gaussians

We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, ...
6
votes
1answer
61 views

Summability of ratios of moments a weight

Recently, I encounter the following problem: Let $w$ be a probability density on $[0,1]$. Let mk be the $k$-th moment, i.e., $$m_k=\int_0^1t^kw(t)dt.$$ Under what condition can we have ...
0
votes
0answers
84 views

An inequality for moments of a random variable

I'm interested in a class C of $R^1$-valued random variables $\xi$ which satisfy an inequality of the type $$ (1) \qquad E|\xi|^p \leq F(E|\xi|^2), $$ where $p>2$, $F$ is a certain ...
0
votes
0answers
57 views

Bounds or approximations for the conditional probability of an event involving correlated random variables

Let $\tilde{\gamma_1}, \tilde{\gamma_2}, \ldots, \tilde{\gamma_N}$ be exponential random variables (RVs) that are correlated with each other. Let $\gamma_n$ be another exponential RV that is ...
7
votes
0answers
255 views

Inequality between incomplete beta and gamma functions; or when is binomial distribution function above/below its limiting Poisson

Please note: this question was posted first (September 4) in math.stackeschange.com and then (September 16) in stats.stackeschange.com. It got no answers in neither of those sites. Let the ...
1
vote
1answer
157 views

Distance between the product of marginal distributions and the joint distribution

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= ...
13
votes
4answers
897 views

A Normal Distribution Inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the following ...
2
votes
0answers
54 views

Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables

Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors: \begin{eqnarray*} \bar{a} &=& ...
7
votes
2answers
320 views

Entropy conjecture for distributions over $\mathbb{Z}_n$

Suppose we have two independent random variables $X$ (with distribution $p_X$) and $Y$ (with distribution $p_Y$) which take values in the cyclic group $\mathbb{Z}_n$. Let $Z = X +Y$, where the ...
2
votes
0answers
190 views

Proving that an increasing iterative sequence increases at a decreasing rate

In this question Proving a sequence of integrals increases (iterated minimax distributions) Pietro Majer proved that $$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = ...
4
votes
2answers
602 views

Proving a sequence of integrals increases (iterated minimax distributions)

I am trying to show that $$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = (1-(1-F_{n-1}(x))^c)^c$$ and $F_0(x) = x$ and $n$ and $c$ are integers, $n\geq 1$ and $c \geq 2$ Note that ...
4
votes
1answer
415 views

total variation distance of product of measures

Let $f, \hat{f}, g,$ and $\hat{g}$ be continuous probability densities. Define probability densities $p \propto fg$ and $\hat{p} \propto \hat{f}\hat{g}$. Is it true that \begin{align*} ||p - ...