0
votes
1answer
33 views

Putting a bound on some probability inequality

Assume that we have the following polynomial: $$ax^2 + bx =c$$ and a, b, c are i.i.d uniform random variables in [0, 1]. I'm trying to calculate the probability that the root is real, and that ...
0
votes
1answer
14 views

Show $P(h(X)\ge a)\ge\frac{E(h(X))-a}{b-a}$

Show that, if $h:\mathbb R\to[0,b]$ and $0\le a< b$ then, $\displaystyle P(h(X)\ge a)\ge\frac{E(h(X))-a}{b-a}$ So $h$ is nonnegative and bounded. If $a=0$ then the inequality holds. because ...
1
vote
1answer
30 views

DKW-style $\ell_{\infty}$ bounds for sum of i.i.d. random functions: $\to [0,1]$

Let $\mathbf{G}$ be the set of (edit: convex) functions $g: X \to [0,1]$, where $X$ is a compact subset of $\mathbb{R}^d$ or something like that. Suppose I have a distribution $D$ on $\mathbf{G}$. ...
2
votes
1answer
29 views

probability theory proof of exponential chebyshev inequality

This is a question about my homework. I am not sure about what is exponential Chebyshev inequality, also how do I get rid of the absolute value and prove it directly by PDF? As well as the ...
1
vote
1answer
49 views

Generalized inverse of the cdf applied to a random variable equals the random variable itself almost surely?

first of all I apologize for the awful title but I really did not know how to formulate a precise question. Consider the following setup. Let $F$ be the distribution function of a random variable ...
1
vote
1answer
24 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 ...
1
vote
0answers
39 views

One-Sided Bivariate Chebyshev Inequality

Let $X$ and $Y$ be random variables with finite means $\mu_X$ and $\mu_Y,$ finite variances $\sigma_X^2$ and $\sigma_Y^2,$ and correlation $\rho.$ Let $A$ be the event that $X \leq \mu_X + k\sigma_X$ ...
1
vote
0answers
32 views

Proving that $ (\mathbb E [X^n])^{1/n}\leq (\mathbb E [X^m])^{1/m}$ for $1\leq X\leq 2 \ \mathbb P \text{-a.e.}$

How to prove that for a positive essentially bounded random variable $X$ satisfing $1\leq X\leq 2 \ \mathbb P \text{-a.e.}$ and for any $m,n \in \mathbb N^*$ with $m\geq n$ we have $$ (\mathbb E ...
2
votes
1answer
56 views

Below bound of the mesure of a finite intersection

Let $(X, \mathcal{M}, \mu)$ be a measure space, with $\mu(X)=1$. If $A_{1}, A_{2}, ..., A_{n} \in \mathcal{M}$, prove that $$\mu \left(\bigcap_{j=1}^{n} A_{j} \right) \geq \sum_{j=1}^{n} \mu{(A_{j})} ...
0
votes
2answers
86 views

Markov's Inequality, only non-negative random variables

I have a question about a Markov's inequality, which states following. Let $X : \Omega \rightarrow \mathbb{R}$ be a non-negative random variable on probability space $(\Omega, \mathscr{A}, P)$ and ...
1
vote
0answers
32 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 ...
4
votes
2answers
71 views

Typo on Wikipedia's entry on Hoeffding's inequality?

In Wikipedia's entry on Hoeffding's inequality, they state that if $\overline{X} := \frac 1 n \sum_{i=1}^n X_i$, then $$ P(\overline{X}-E[\overline{X}] \ge t) \le \exp (-2n^2 t^2)$$ if we assume for ...
1
vote
0answers
41 views

Prove or disprove an inequality involving statistics

Do we have any result in statistics like this: $$|\overline x - \mu_e| \leq \sigma$$ Here $\overline x$ denotes the usual mean of some given discrete observations, $\mu_e$ their median and ...
0
votes
0answers
111 views

Proof of Mrs. Gerber's Lemma Using Convexity and Jensen's Inequality

Can anyone give a proof of the Mrs. Gerber’s Lemma for the scalar case: $$ \ H^{-1}(H(Y|U)) \ge H^{-1}(H(X|U))*p $$ where $$ \ a*b = a(1-b) + b(1-a) $$ $$ \ X\ ,\ Y $$ are binary random ...
1
vote
0answers
66 views

Sequence of probabilities with monotone function

Let $\{X_k\}_{k=1}^{\infty}$ be a sequence of i.i.d. random variables with finite support $S = \{ 1, 2, ..., N\}$. Let $P$ be the corresponding probability measure. For all $k \geq 1$, define $A_k := ...
0
votes
1answer
62 views

Inequality with monotone functions on power set

Consider a discrete probability space $\left( S, F, P\right)$, where $S = \{ 1, 2, \ldots, N \}$. Consider the set $$S' := \mathcal{P}(S) \setminus \{ \varnothing\} = \{ \{ 1\}, \{ 2\}, \ldots, ...
0
votes
1answer
54 views

Which different probabilistic bounds/inequalities apply when we are given a lower bound on the sample size

Let m be the sample size and $X_i$ be a r.v. that we sample and define a new r.v. such that: $$M_m=\frac{1}{m}\sum^m_{i=1}{X_i}$$ My question is, what type of probabilistic inequalities require some ...
1
vote
0answers
103 views

How to use Chebyshev's inequality or the law of large numbers to a probability question

Let x be a random bit string that takes values $\{1,0\}^n$. Let r be the value of the most significant (MSB) bit of x (and r is a r.v. 1 or 0 that are equally likely). Let g be our guess for the MSB ...
2
votes
3answers
153 views

Inequality for Expected Value of Product

Let $(\Omega, \mathbb{P}, \mathcal{F})$ be a probability space, and let $\mathbb{E}$ denote the expected value operator. Consider the random variables $f: \Omega \rightarrow \{0,1,2\}$ and $g: \Omega ...
1
vote
1answer
68 views

Let $Z$ be a stochast with $EZ = 0$ and $VarZ = \sigma^{2}$. Show that for $u,v>0$ that the following inequality holds:

Let $Z$ be a stochast with $EZ = 0$ and $VarZ = \sigma^{2}$. Show that for $u,v>0$ that the following inequality holds: $P(Z\leq -u \space \text{or} \space Z\geq v) \leq \frac{4\sigma^{2} + ...
2
votes
1answer
57 views

Hoeffding-type bound covering all partial sums? (better than naive union bound)

I think this must have been done before, possibly with martingales, but I can't find anything online! Given $X_1,\dots,X_n$ independent, each $X_i \in [a_i,b_i]$, letting $S_i = \sum_{j=1}^i X_j$, do ...
0
votes
0answers
24 views

Problem with an inequality from probability theory (Random matrix theory)

I read the following notes on random matrix theory http://www.umpa.ens-lyon.fr/~aguionne/cours.pdf . While reading Wigner's proof for the semi-cicular law I encoutered the following inequality on page ...
0
votes
1answer
40 views

Impact of variance on the characterization of random variables

Suppose I have two discrete random variables $X$ and $Y$, and all I know is that they take values from the same set of non-negative integers, their expectations are the same ...
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. ...
2
votes
1answer
54 views

Chebyshev-like lower bound

Using the well-known Chebyshev's inequality we can upper-bound the tail of the distribution of a random variable $X$ using its variance $\sigma^2_X$ moment as follows: $$P(|X-\mu_X|\geq ...
1
vote
1answer
19 views

Inequality involving generalized harmonic numbers

I am working though YS Chow's Probability Theory and have found an equality that I cannot justify. In theorem 3 on page 118 the following inequality is used: $\sum_{n=j}^\infty \frac{1}{n^2} \le ...
5
votes
1answer
91 views

Hoeffding inequality adapted to discrete random variables

Given $n$ (real-valued) random variables $X_1, X_2, ..., X_n \in [0, B]$, it can be derived from Hoeffding's Inequality that: $$\mathbb{P}^n\left[ \bar{X} - \mathbb{E}_n[ \bar{X} ] \geq t \right] \leq ...
3
votes
0answers
56 views

Is Hoeffding's bound tight in any way?

The inequality: $$\Pr(\overline X - \mathrm{E}[\overline X] \geq t) \leq \exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right) $$ Is this bound (or any other form of hoeffding) tight in ...
-1
votes
1answer
28 views

Inequality - Normal Distribuion

I have the next inequality that I need to prove: If $X$ has a Normal $(\mu,\sigma^{2})$ distribution. $$P(|X-\mu|>\varepsilon\space\sigma)\le ...
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 ...
2
votes
1answer
38 views

Problem on inequality

Prove that, $E|X|^p < \infty $ iff $\sum_{k=1}^{\infty}k^{p-1}P\{|X| \geq k\} < \infty$. Where E is the expectation and P is the usual probability measure. There was a mistake one it's correct.
1
vote
1answer
48 views

Bounding the integral of the tails of a random variable.

I found an argument like this in a book, but I couldn't understand how we got this bound. Suppose $X_n$ is a sequence of random variables. For some $\delta > 0$ and all $n \geq 1$, $$ \int_{|X_n| ...
2
votes
2answers
94 views

Strange deduction about relation of median and mean

On his blog T. Tao's proves the following concentration inequality, due to Talagrand. Let $K>0$, and let $X_{1},..., X_{n}$ be iid complex random variables all bounded by $K$. Let ...
1
vote
1answer
151 views

Distance between mean and median

I want to solve the following problem in T.Tao's random matrix theory book. Let $X$ be a random variable with finite second momment. A median $M(X)$ of $X$ saisfies ...
0
votes
1answer
86 views

Application of Markov's inequality

let $h\colon \mathbb{R} \to [0,\alpha]$ be a nonnegative bounded function. Show that for $0\leq a<\alpha$ that the following holds: \begin{equation} Pr(h(X)\leq a) \geq \frac{E[h(X)]-a}{\alpha-a} ...
1
vote
1answer
105 views

Lower bound of Fourier transform

We know the Fourier transform of the Gauss-function: $\displaystyle\int_{\xi\in\mathbb{R}^d}e^{-\pi\, C\,|\xi|^2}e^{2\pi i \xi\cdot X}d\xi=C^{-d/2}e^{-\, \pi\, |X|^2/2}$ for any $C>0$. Then ...
2
votes
1answer
75 views

Esséen concentration inequality

I want to prove the following: Let $X$ be a random variable taking values in $\mathbb{R}^d$. Then for any $r>0, \epsilon >0$: $\displaystyle \sup_{x_{0}\in \mathbb{R}^d}{\bf{P}}(|X-x_{0}|\leq ...
0
votes
1answer
107 views

Lower bound on the probability of maximum of $n$ i.i.d. chi-square random variables exceeding a value close to their number of degrees of freedom

I am wondering if there is a tight lower bound on the probability of a maximum of $n$ i.i.d. chi-square random variables, each with degree of freedom $d$ exceeding a value close to $d$. Formally, I ...
3
votes
1answer
258 views

Average absolute value of sum with Rademacher random variables

Let $a_1, \ldots, a_n $ be independent Rademacher random variables with distribution $P(a_i=1) = P(a_i=-1) = \frac 12$. Estimate from below $$E \left|\sum_{i=1}^n a_i\right|.$$ I've reduced this ...
2
votes
0answers
70 views

Metric on the set of CDFs with finite p-th moment

Let $\mathcal{F}_p$, $p \ge 1$, be the set of all cumulative distribution functions of real valued random variables whose $p$-th moment is finite. I'm looking for a metric on $\mathcal{F}_p$ and ...
1
vote
0answers
101 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 ...
1
vote
1answer
78 views

How to prove this simple inequality?

Please help me to prove this inequality. Suppose $X$ and $Y$ are independent and $EX=EY=0$, then we must have $E(|X|) \leq E(|X+Y|)$. Thanks.
6
votes
0answers
370 views

Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
3
votes
0answers
97 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} ...
2
votes
2answers
77 views

$ E\left( \left|\frac{1}{n}\sum_{j=1}^n X_j\right|^p \right) \le \left( \frac{1}{n}\sum_{j=1}^n E(|X_j|^p)^{1/p} \right)^p$

The following is problem 14 of section 3.2 from Chung's "A Course in Probability Theory". If $p>1$, we have $$\left| \frac{1}{n}\sum_{j=1}^n X_j \right|^{p} \le \frac{1}{n}\sum_{j=1}^n |X_j|^p$$ ...
4
votes
1answer
62 views

$P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$

If $E(X^2)=1$ and $E(|X|)\ge a >0$, then $P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$. I can see from the well known inequality $E(|X|) \le E(|X|^2)^{1/2}$ that it must be the ...
3
votes
0answers
88 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 $X$, $Y$ ...
3
votes
0answers
102 views

a simpler proof for an inequality in probability

I am trying to show that for any two real $X,Y$ iid there holds that $$ P(|X - Y| \le 2) \leq 3P(|X - Y| \le 1) $$ I am getting nowhere with this and so I did some digging and found the following ...
2
votes
1answer
39 views

Variance inequality to show deviation from midpoint

How to show this inequality: If $\mathbb P (X \in [a,b]) = 1$, then $\operatorname{Var}(X) \leqslant \frac{(a-b)^2}{4}$. Thank you!
2
votes
0answers
71 views

Probability Theory: How to prove inequality $\mathbb{E}|X - m|^3 \leq \mathbb{E}|X|^3 (1 + \frac{m}{\sigma})^3$

Let's define $X$ - random variable with $F(x)$ distribution function. Also, denote $m = \mathbb{E}X$ and $\sigma^2 = \mathbb{D}X$. Suppose, that $m>0$. How to prove this inequality in these ...