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24 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 ...
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1answer
25 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 ...
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0answers
28 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 $$ ...
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0answers
18 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 ...
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1answer
67 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 ...
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1answer
73 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 ...
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0answers
201 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
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0answers
38 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
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3answers
298 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
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1answer
104 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
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4answers
379 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?
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0answers
19 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
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1answer
55 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
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0answers
33 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
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1answer
76 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 $X \leq \mu_X + k\sigma_X$ ...
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1answer
84 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 ...
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2answers
499 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 ...
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1answer
74 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
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1answer
114 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.
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0answers
54 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
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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 ...
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1answer
69 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 $0.25$.
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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 ...
2
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1answer
159 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 ...
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1answer
342 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
61 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
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1answer
208 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
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1answer
61 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. ...
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0answers
75 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 $x$ ...
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1answer
116 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 ...
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0answers
36 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 $t\in[a,b]$, ...
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0answers
51 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 ...
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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 ...
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1answer
436 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
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1answer
295 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 ...
0
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1answer
98 views

Why does KL divergence show up in the proof of Hoeffding's inequality?

In some textbook the KL divergence shows up in the proof of Hoeffding's inequality (e.g., eq. (5) of this material). In contrary, most other textbooks seem not mention this fact. I know that KL ...
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0answers
64 views

Why Hoeffding's inequality has a sharper bound than Markov's inequality?

Basically, I can understand the proofs' details for both inequalities, but still I have no idea why the bound of Hoeffding is sharper than that of Markov? Is there any underlying intuitive that can be ...
0
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1answer
91 views

Berry-Esseen theorem

In the Berry-Essen theorem, why is the standard normal distribution used in the context of the closedness of two distributions? Why can't we use the general normal distribution where will things will ...
4
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1answer
140 views

What can we conclude about the distribution of the sum of two random variables?

If we know, for independent random variables $X$ and $Y$, $P(X>x)\leq0.05$, and $P(Y>y)\leq0.05$, can we say anything about $P(X+Y>x+y)$? Can we be certain that it is less than $0.05$? Under ...
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0answers
54 views

About Galton watson process

My question is about a homework question that I found interesting. It gives another proof (without using martingales) for that the critical Galton Watson tree dies out eventually. But it has given a ...
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0answers
122 views

About tail distribution of a sum

Do we know anything about the tail distribution of sum of squares of a limited number of i.i.d exponentially distributed random variables? I'm looking for a good bound.
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2answers
168 views

Where can I use Chebychev's inequality?

What are some real-world applications where I can leverage Chebychev's inqueality? All the examples I find are either related to coin tosses or some school scores related problems. Are there any ...
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0answers
348 views

Tail inequality on sum of product of normal variables

For independent random variables $ x_1,..,x_n$ and $y_1,...,y_n$ following normal distribution $N(0,1)$, I need a simple estimate formula for $P(| \sum_1^n x_iy_i | \leq nt ) \leq e^{(?)}$ for ...
4
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1answer
229 views

Inequality for bivariate normal distribution

Let $X_1$ and $X_2$ be bivariate normal with mean $\mu=(0,\mu_2)$, for any $\mu_2$, and correlation $\rho$. Consider the following inequality: \begin{align*} Pr\left\{|X_1| \ge ...
3
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1answer
272 views

Order statistics of absolute value of bivariate normal distribution

Suppose $X_1$ and $X_2$ are bivariate normal and let $|X|_{(1)}$ and $|X|_{(2)}$ be the ordered version of their absolute value. I am interesting in finding the following probabilities or some bounds ...
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1answer
1k views

How to compare or normalize two sets of probabilities?

I'm a stats newbie so I hope this question isn't too easy for you! :) I'm not sure how to ask it so I've written two versions: the short version is a pure stats question and the longer version ...
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0answers
17 views

Inequality distribution function [duplicate]

Possible Duplicate: Inequality distribution function I am trying to understand a proof (no homework). If $\frac{\overline{F}(x-y)}{\overline{F}(x)} \rightarrow 1$ for $x \rightarrow ...
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1answer
169 views

Inequality convolution distribution function

$F$ is the distribution function of a positive random variable. Why does the following hold for every $x \in (0,\infty)$: $F^{*2}(x) \leq F^2(x)$, where $F^{*2}=F*F=\int\limits_0^x F(x-y)dF(y)$
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1answer
294 views

An instance of the Cauchy–Schwarz inequality

In the proof of Theorem 6.5 from the book by Devroye et al., how is the last inequality derived? $$ \begin{aligned} \mathbb{E}\left\{|\eta(X)-1/2|\mathbf{I}_{\{g(X)\ne g^*(X)\}}\right\} &\leq ...
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1answer
335 views

Bounded in probability ($o_{P}$)

In Larry Wasserman's lecture notes on $o_{P}$ and $O_{P}$, I am not able to follow the derivation of the following example in page 9. Consider $m$ coins with probabilities $p_{1}, \ldots ,p_{m}$. ...