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1
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1answer
15 views

Inequalities for maxima over random functions

Let R be the set of real numbers. Say we have a function $f(x,X) \in R $ where $X \in R^d$ is a random variable over $\Omega$ and $x \in R$. I'm searching for an upper bound for the expected value of ...
11
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3answers
122 views

Regarding convergence in probability

Let $\{X_n\}_{n\geq 1}$ be a sequence of random variables s.t $X_n \to a$ in probability, where $a>0$ is a fixed constant. I'm trying to show the following: $$\sqrt{X_n} \to \sqrt{a}$$ and ...
2
votes
0answers
25 views

upper bound for expected maximum of difference of two kernel-Estimations

I'm searching for an upper bound for a function like $$ E\left[ \max_{x \in R} \left( \frac{ \sum_{i=1}^n K(\frac{x-X_i}{f(x,X_1, \dots X_n)}) \cdot Y_i } { \sum_{i=1}^n ...
2
votes
2answers
107 views

Moving an expectation inside a probability?

Let $X$ and $Y$ be two independent random vectors such that $E[X^TY]>0$, and all components of $X$ are positive valued. Then, is it true that, $P_{X,Y}\{X^T Y > 0\} \le P_{Y}\{E_X[X]^T Y > ...
5
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0answers
106 views

A question related to Borel-Cantelli Lemma

Note: Borel-Cantelli Lemma says that $$\sum_{n=1}^\infty P(A_n) \lt \infty \Rightarrow P(\lim\sup A_n)=0$$ $$\sum_{n=1}^\infty P(A_n) =\infty \textrm{ and } ...
1
vote
1answer
34 views

How to combine probability inequalities that are w.r.t. different random variables?

Let $x$ and $z$ be two independent random variables. Suppose I know the following two facts: $P_z[f(x,z) < g(x)] > 1-\delta$ uniformly for all x; $P_x[g(x) < h(x)] > 1-\delta_h$ How can ...
2
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0answers
38 views

Probability inequality implication

Let $X$ and $Y$ be independent random variables, and $A(X,Y)$ a predicate with values true or false. Suppose we know the following: $P_Y[\forall X, A(X,Y) \Rightarrow B(X)] > 1-\delta$. Can we ...
3
votes
2answers
97 views

$E(X)E(1/X) \leq (a + b)^2 / 4ab$

I've worked on the following problem and have a solution (included below), but I would like to know if there are any other solutions to this problem, especially more elegant solutions that apply well ...
1
vote
1answer
43 views

Chi-square test on a sample from population having an unequal distribution

I got a sample with an unequal gender distribution ($30$ males to $200$ female). This distribution is quite "representative" for the population the sample is taken from. Nevertheless, in a paper, ...
1
vote
0answers
21 views

Results stronger than Dvoretzky–Kiefer–Wolfowitz inequality?

There is the DKW inequality which controls the extent to which the empirical cdf of a sample from a real-valued random variable differs from the true cdf. Are there any stronger results (i.e. ...
2
votes
0answers
29 views

Is this a true inequality over multinomial parameters?

Let $p_1,\ldots,p_k$ be the multinomial parameters sampled from a Dirichlet distribution. Assume $\bar{p_1},\ldots,\bar{p}_k$ be the mean of Dirichlet distribution. Then, $$Pr( ...
0
votes
0answers
15 views

Getting a “friendly” tailbound from a closed-form description of the probability density (the case of the n-th order statistic)

Suppose I have a probability distribution of an $n$-th order statistic $X_n$ with mean $\mu$ and density $f_n(x)$, where $n$ scales to infinity. If one wants a concrete example, the one I care about ...
2
votes
0answers
18 views

Identifying values responsible for producing inequality or concentration within a distribution

(This may be a dumb question I simply don't know enough to know is dumb, but it's foxing a surprisingly large chunk of my social circle) I have a dataset web requests - specifically, the page, and ...
1
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0answers
27 views

Tail bound on dependent sum

Let R be a random variable, and let X_i be a nonlinear, nonconvex function that takes R and outputs a value in [-1,1]. I'm trying to get a tail bound on the following dependent sum: ...
3
votes
1answer
68 views

Understanding a step of a proof of Markov's inequality

$X$ is a positive valued random variable, $\theta$ is greater than $1$, $P$ is a probability measure. Why is the following inequality true: $$\int_{\theta E(X)}^{+\infty} x dP_{X}(x) \ge ...
1
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0answers
26 views

Families of distributions with bounded variations

For what family of probability distributions $f(x)$ do we have the following property? $$ \forall x, \quad \int f(u) - f(u+x) du \leq L\| x \| $$ for some $L$. Can we say anything about the ...
1
vote
1answer
70 views

Tail bounds for Beta distribution

Say $X\sim\mathrm{Beta}(\alpha,\beta)$. Are there any "nice" closed form upper bounds for the tail probability $P(X\geq\epsilon)$, that are reasonably tight when $\beta$ is large? By "nice" I mean ...
0
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0answers
30 views

how can Cauchy-Schwarz help here?

From http://math.stackexchange.com/questions/1004544/if-mathrme-x2-exists-then-mathrme-x-also-exists If $\mathrm{E} |X|^2$ is finite, then $\mathrm{E} X$ exists and is finite, because $$ ...
0
votes
0answers
25 views

I don't understand the solution to this Chernoff inequality?

I have a sum, $S_n = \sum_{i=1}^n X_i$ of n iid Poisson distributed random variables $X_1,...X_n$ I am supposed to apply the Chernoff bound to $S_n$. My professor gave us the solution: However, I ...
5
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1answer
164 views

How to bound a probability with Chernoff's inequality?

In my class, we were given Chernoff's inequality as $$P(X\le -t) \le e^{(-(\lambda t - \log( E(e^{-\lambda x}))))}$$ $$P(X\ge -t) \le e^{(-(\lambda t - \log( E(e^{\lambda x}))))}$$ It says that to ...
2
votes
1answer
109 views

Does vector version of the Cauchy-Schwarz inequality ensure that the correlation coefficient is bounded by 1?

I have been trying to understand the proof that the correlation between two random variables $X$ and $Y$ is between $-1$ and $1$. For simplicity, suppose $X$ and $Y$ have mean zero. Then ...
4
votes
0answers
392 views

Cantelli's inequality proof

I am trying to prove the following inequality: EDIT: Almost immediately after I posted this question, I discovered that the inequality I am being asked to prove is called Cantelli's inequality. When ...
2
votes
0answers
45 views

Concentration inequality of weighted sum of random variables given a tail inequality

I'm reading this book on concentration inequalities and I'm trying to solve all of the exercises in the book. The following problem is from the book which I couldn't manage to solve. I have also ...
9
votes
3answers
327 views

Exponential upper bound

Suppose we have IID random variables $X_1,\dots,X_n$ with distribution $\mathrm{Ber}(\theta)$. We are going to observe a sample of the $X_i$'s in the following way: let $Y_1,\dots,Y_n$ be independent ...
1
vote
1answer
115 views

Proof of density inequality

I was wondering if there is an easy way to find sufficient conditions for the following inequality to hold $$ \int f(x,y)^2 \:\mathrm{d}x \:\mathrm{d}y - \int f(x)^2 f(y)^2 \:\mathrm{d}x\:\mathrm{d}y ...
3
votes
4answers
390 views

Given more information, can a probability lessen?

Let $A$, $B$ and $C$ be events in the same probability space. Does $$\begin{align} \mathbb P(A\,|\,B\cup C) \ge \mathbb P(A\,|\,B) \end{align}$$ hold?
1
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0answers
23 views

Need help on Cramér-Chernoff method in concentration inequalities

I am reading up on the Cramér-Chernoff method in concentration inequalities. The idea is to use Markov's inequality and the monotonic transformation $\phi(t) = e^{\lambda t}$ where $\lambda \geq 0$. ...
0
votes
1answer
56 views

Probability of ($x\le 2Y$)

$X$ and $Y$ are independent and their probability density functions are $$f_X(t)=f_Y(t)=\left\{\begin{array}{l} e^{-t},\:\text{if $t \geq 0$;} \\ 0,\:\text{otherwise.}\end{array}\right.$$ $P(X \leq ...
2
votes
0answers
44 views

$\sigma^2 \le (\mu-a)(b-\mu)$ for all probability distributions bounded on $[a,b]$? [duplicate]

Let $\mu$ be the mean and $\sigma$ the standard deviation of a probability distribution defined on the bounded interval $[a,b]$ (that is, the probability that the random variable lies outside $[a,b]$ ...
3
votes
1answer
97 views

Bivariate One-Sided Chebyshev Inequality (Symmetric Case)

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
1answer
116 views

Randomized Response, calculating probability

I have a problem with randomized response method when calculating the probability of critical attribute. For any of you who are not familiar with the method, you can check it here, Wikipedia Entry for ...
21
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2answers
619 views

Does a sample version of the one-sided Chebyshev inequality exist?

I am interested in the following one-sided Chebyshev inequality: $$ \mathbb P(X - \mathbb E (X) \geq t) \leq \frac{\mathrm{Var}(X)}{\mathrm{Var}(X) + t^2} \,. $$ Basically, if you know the ...
0
votes
1answer
83 views

Variance Inequality for Random Vectors

I know that if X and Y are random scalar variables, then: \begin{align*} \mathrm{Var}(X+Y) & = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2\mathrm{Corr}(X,Y)\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)} \\ ...
3
votes
1answer
120 views

Is it true that $E[e^{tX}] \le e^{E[t^2X^2/2]}$?

I've seen some one use the inequality $E_X[e^{tX}] \le e^{E_X[t^2X^2/2]}$. However, I don't know if it is true and can not prove it. Thanks for help.
2
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0answers
63 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 ...
2
votes
0answers
60 views

An inequality involving expectation

Let $f,g$ be two pdfs, and suppose $X$ is a random variable that has pdf $f$. Is it necessarily true that $E[f(X)] \ge E[g(X)]$? Although I doubt this will help, but I got this problem from studying ...
4
votes
1answer
74 views

If Chebyshev's upper bound gives same value as the actual probability calculation, what can we conclude?

As an example, if Chebyshev tells $P(|X-\mu|\geq k\sigma)\leq 0.25$ and the actual probability for $k=2$ is also $0.25$.
5
votes
2answers
232 views

Concentration bounds on a sequence of (0,1)-experiments

I have a $(0,1)$-experiment that returns $1$ with probability $p$ and $0$ with probability $1-p$. Let $X_i$ be the random variable describing the outcome of iteration $i$ of the experiment. I'm ...
3
votes
1answer
195 views

Can the mean deviation about mean exceed the standard deviation for the Pareto distribution?

Can the mean deviation about mean exceed the standard deviation for the Pareto distribution? I just went through some books and found they are claiming that it cannot. How can I prove that? What is ...
20
votes
1answer
446 views

When is binomial distribution function above/below its limiting Poisson distribution function?

Let $B(n,p,r)$ denote the binomial distribution function (DF) with parameters $n \in \mathbb N$ and $p \in (0,1)$ evaluated at $r \in \{0,1,\ldots,n\}$: \begin{equation} B(n,p,r) = \sum_{i=0}^r ...
2
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0answers
68 views

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

[I have cross-posted this from math.stackexchange: http://math.stackexchange.com/questions/476466/is-the-following-property-for-positive-random-variables-fulfilled-in-general ] Suppose we have a ...
9
votes
1answer
215 views

Understanding measure concentration inequalities

In the spirit of this question Understanding proof of a lemma used in Hoeffding inequality , I am trying to understand the steps that lead to Hoeffding's inequality. What holds the most mystery for ...
3
votes
1answer
68 views

Relations between probabilities of “almost” independent random variables

Let $X$ and $Y$ be two random variables, such that the (average) mutual information is very small: $$ 0 \le I(X;Y) \le \epsilon \ll 1$$ In this case, we say that $X$ and $Y$ are almost independent. ...
2
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0answers
76 views

Bounding expected maximum of inner products

Let $x=(x_{1},\dots,x_{n})^{T}$ be $n$ dimensional random vector uniformly distributed over the $L1$ unit sphere. That is, all $x$ such that $\|x\|_{1} = 1$ have equal probability and all other $x$ ...
9
votes
1answer
124 views

What makes Hoeffding's inequality an important statistical concept?

On his blog, Larry Wasserman has a post about what he planned to cover in his course last fall. He notes that he was abandoning some classical topics in favor of more modern issues. One topic that he ...
4
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0answers
40 views

Index of dispersion with approximate distribution

I have an unknown discrete probability distribution $D$ ($D$ is a probability mass function), defined on an interval $[a,b]$ ($a>0$) and an estimation $\hat{D}$ such that, for all $t\in[a,b]$, ...
0
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0answers
52 views

There is a load pdf and a capacity pdf, what is pdf of the surplus (the difference)?

I have a probability distribution function for expected available energy $\text{P}(t)$ and another pdf for expected load $\text{E}(t)$. How do I find a pdf for the set of random variables of ...
1
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0answers
23 views

Correction term in variance of mixture [duplicate]

In the answer to this question, the correction term is claimed to be nonnegative: $$p_A \mu_A^2 + p_B \mu_B^2 - p_A^2 \mu_A^2 - p_B^2 \mu_B^2 - 2 p_A p_B \mu_A \mu_B \geq 0$$ where $p_A + p_B = 1$, $0 ...
3
votes
1answer
557 views

Is Jensen's inequality applicable for two variables?

This is the Jensen's inequality I saw in my textbook: $$E{ f(X) } \geq f( E(X) ),$$ where $f$ is a convex function. Is this also applicable for two random variables--independent or otherwise--like ...
1
vote
1answer
374 views

Intuition behind Dvoretzky Kiefer Wolfowitz inequality

I've been reading about the Dvoretzky-Kiefer-Wolfowitz inequality, in the context of confidence bands on empirical distribution functions. I think I understand the inequality at face value: that the ...