0
votes
0answers
16 views

Vysochanskij Petunin vs. Cantelli inequality for random variables

The well known Cantelli inequality states: $$Pr(|X-\mu|\ge\alpha)\le\frac{2\sigma^2}{\sigma^2+\alpha^2}$$ where $X$ is a real valued random variable, $\mu$ the mean value and $\sigma^2$ the variance ...
0
votes
1answer
36 views

Approximation in Normal distribution random variable

Let ${X_n : n \geq 1}$ be independent $\mathcal{N}(0,1)$ random variables. How do we get the following approximation?
0
votes
0answers
42 views

Help with an inequality of probability distribution functions

There are six random variables $X_{1}$, $X_{2}$, $X_{3}$, $Y_{1}$, $Y_{2}$, and $Y_{3}$ on $[0,c]$. Their cumulative distribution functions are $% F_{1}(t)$, $F_{2}(t)$, $F_{3}(t)$, $G_{1}(t)$, ...
1
vote
1answer
22 views

Two Uniform Independent Random Variables: When is one greater?

You have two independent random variables: $X$ and $Y$, which are both uniformly distributed over $(0,1)$. Consider the inequality $X^2- 4Y < 0$. What percentage of the time is the inequality ...
0
votes
1answer
33 views

Error Term of Chebyshev inequality?

Chebyshev inequality tells us that $$Pr[|X-E[X]|\geq a]\leq \frac{Var[X]^2}{a^2}$$ Do you know an Expression (or a paper where this Expression is mentioned) for the error term?
1
vote
0answers
30 views

Showing an inequality relating two Poisson tail-probabilities

In my research, I've discovered that a property that I am interested in is equivalent to an inequality involving two tail-probabilities of the Poisson distribution. I belive this inequality to be ...
0
votes
1answer
58 views

Bias of expected binomial reciprocal

$X$ ~ Bionomial$(n, p)$. I want to evaluate the bias of $E\left(\frac{1}{X}\right)$,assuming $X>0$ (excluding the probability $(1-p)^n$), that is, the difference between $E\left(\frac{1}{X}\right)$ ...
0
votes
0answers
35 views

pinsker's inequality

I was wondering if someone knew or could explain what the pinsker's inequality means from a probability theory point of view. I know the mathematical formulation but dont quite get the essence. ...
1
vote
1answer
68 views

$\int_{t=-\infty}^x (G(t)-F(t))\mbox{d}t\geq 0\forall x$ and $\frac{\mbox{d}F(t)}{\mbox{d}G(t)}$ increasing $\Longrightarrow G(x)\geq F(x)\forall x$?

As in the title I wonder if the relation at the left side implies the one at the right side. Better to rewrite it clearly Given are: $\mbox{Info} (1)\rightarrow$ $\int_{t=-\infty}^x ...
3
votes
2answers
168 views

Norm of random vector plus constant

Suppose that $w$ is a multivariate standard normal vector and $c$ a real vector of the same size. I know that for positive x $$P(||w+c||^2\geq x)\ \geq \ P(||w||^2\geq x)$$ but I cannot prove it. We ...
0
votes
0answers
24 views

Finding a (tighter) sufficient condition on the standard deviation of a random variable

Let $\tilde{\phi}$ be a non-negative random variable with a mean normalized to $1$, with $F(\phi) := \Pr(\tilde{\phi} \leq \phi)$ denoting its CDF. $F(\phi)$ is assumed to be twice continuousy ...
0
votes
0answers
28 views

Bounding the standard deviation of a random variable

I have the following problem. Let $\tilde{\theta}$ be a non-negative random variable with twice continuously differentiable cdf $F(\theta) := \Pr(\tilde{\theta} \leq \theta)$ and $E(\tilde{\theta}) = ...
1
vote
1answer
125 views

How prove this distributions inequality $cov(\theta_{i},\theta_{j})\ge 0$?

Question: let random variable $\theta$ has dendity $f_{\phi}(\phi)$,and the random vector $\theta=(\theta_{1},\theta_{2},\cdots,\theta_{n})$,such $\theta_{i}|\phi$ are all independent from each ...
1
vote
1answer
69 views

Tail probability of the $\chi^2$ distribution

Ho to prove that $$ \int_{2s\epsilon^{-2}}^{\infty}\frac{1}{\Gamma(d/2)2^{d/2}}x^{d/2-1}e^{-x/2}dx \leq const.\epsilon^{-d}\exp(-\epsilon^{-2}s) $$ holds for $\epsilon >0$ sufficiently small? Here ...
2
votes
1answer
77 views

Prove expectation inequality

Any ideas on how I could prove the veracity or falseness of the following inequality? Let $X:\Omega \to \mathbb{R}$ a random variable such that the expressions under are well-defined. Then $$E[e^X] ...
1
vote
1answer
88 views

Is the following property for positive random variables fulfilled in general?

Suppose we have a continuous random variable $X$, defined on the interval $[0, \infty)$, which has density $f(x)$ and a finite expectation and variance. I am wondering whether the following is true ...
1
vote
1answer
67 views

Closed-form expression (or good upper bound) for $\mathbb{E}\left[|X-\mathbb{E}X|^{\alpha}\right]$, where $X$ is binomial?

I am struggling to get either a closed-form expression, or as tight an upperbound as possible, for the quantity $$ M_\alpha(X)\stackrel{\rm{}def}{=} \mathbb{E}\left[|X-\mathbb{E}X|^{\alpha}\right] $$ ...
2
votes
2answers
208 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) &= ...
1
vote
0answers
98 views

Normal distribution inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. Prove the following inequality. $$(x^2+1)N + xn-(xN+n)^2>N^2$$ where the dependency of $n$ and $N$ on ...
2
votes
2answers
129 views

Comparing the relative entropies of some stochastically ordered distributions

Motivation of this question: This question is related to the expected stopping time of a stochastic process under two hypotheses. Especially, it answers the question "how many more samples are ...
3
votes
0answers
96 views

Inequality of covariances between a bivariate normal vector and its indicator functions

Why holds for a standardized bivariate normal vector $Z:=(Z_1,Z_2)$ that \begin{equation} |\operatorname{cov}(Z_1,Z_2)|\geqslant |\operatorname{cov}(1\{Z_1\leq u\},1\{Z_2\leq u\})|? \end{equation} ...
0
votes
1answer
60 views

Boltzmann Distribution With Constraints

I have a problem with showing the existence of Boltzmann distribution given some constraints. Consider $p_1,...,p_n$ a Boltzmann distibution, where $p_i=\frac{\epsilon^{-\beta \cdot E_i}}{\sum_{j}^{} ...
3
votes
0answers
87 views

A question about the stability of a property of the normal distribution

Recall that the standard normal distribution can be characterized as the unique standardized (having mean zero and unit variance) distribution $P$ on $\mathbb{R}$ with the property that with $X$, $Y$ ...
0
votes
0answers
162 views

Distribution of convex combination of i.i.d Gamma random variables

I am wondering what one can say regarding the convex combination of i.i.d Gamma random variables? Specifically, consider $x_{i}$ be $Gamma(\theta,1)$, then would we have the following and if yes, ...
0
votes
0answers
556 views

Binary symmetric channel capacity or mutual information inequality

I proved that I(X,Y) <= 1 - H(p) to the following way: How can I prove if I start in that way I(X,Y) = H(X) - H(X|Y), I ...
1
vote
2answers
81 views

A probability question on sum

Let $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$ and $X_{6}$ be real-valued random variables that have the same probability distribution with finite moments, and they are independent. Does anyone know ...
5
votes
2answers
382 views

Repeatedly rolling a die and the tails of the multinomial distribution.

For $1\leq i\leq n$ let $X_i$ be independent random variables, and let each $X_i$ be the uniform distribution on the set ${0,1,2,\dots,m}$ so that $X_i$ is like an $m+1$ sided die. Let ...
2
votes
2answers
110 views

How can I prove this inequation $\Pr\{X+Y<t\} \le \Pr\{X<t\} \Pr\{Y<t\}$

Could you please help me to prove the inequality probability as follows: $\Pr\{X+Y<t\} \le \Pr\{X<t\} \Pr\{Y<t\}$ where $X$ and $Y$ are non-negative independent random variables with common ...
6
votes
2answers
275 views

Beta Function — finding a lower bound based on parameters

I would like to show that $$ 1-\frac{1}{c}Beta\left(c+1,\frac{1}{c}\right) \geq \frac{1}{c+1}.$$ for all $c \geq 2$. I have plotted it out for $c$ up through 200, and it seems to hold. Does anyone ...
1
vote
1answer
119 views

Does an inequality between definite integrals imply an inequality between the derivative wrt an exponent?

I have two cdfs, both distributed over 0 to 1. Let's call them $F(x)$ and $G(x)$. If I know that $$\int_0^1 F(x) \,dx < \int_0^1 G(x) \,dx$$ then, does it follow that $$ \left|\frac{d}{dn} ...
6
votes
3answers
1k views

Quantile function properties

I am confused by "Inverse distribution function (quantile function)" section of the wikipedia page on CDFs . It says that $$F^{-1}(F(x)) \leq x\text{ and }F(F^{-1}(y)) \geq y$$ However, I ...
5
votes
2answers
4k views

Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$