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4
votes
0answers
46 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
35 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
276 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
103 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
376 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
vote
0answers
15 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
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
votes
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
votes
1answer
55 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
57 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 ...
19
votes
2answers
410 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
61 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
113 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
votes
0answers
47 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
58 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
64 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
229 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
votes
1answer
137 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 ...
14
votes
1answer
247 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
votes
0answers
56 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
195 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
59 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
votes
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 $x$ ...
8
votes
1answer
107 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
votes
0answers
35 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
votes
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 ...
1
vote
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 ...
2
votes
1answer
341 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
232 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
votes
1answer
95 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 ...
2
votes
0answers
59 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
votes
1answer
87 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
votes
1answer
137 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 ...
0
votes
0answers
52 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 ...
4
votes
0answers
115 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.
6
votes
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 ...
5
votes
0answers
309 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
votes
1answer
214 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
votes
1answer
260 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 ...
3
votes
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 ...
0
votes
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 ...
2
votes
1answer
168 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 $F^{*2}=F*F=\int\limits_0^x F(x-y)dF(y)$
2
votes
1answer
287 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 ...
4
votes
1answer
334 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 $p_{1}, \ldots ,p_{m}$. ...
3
votes
1answer
373 views

Understanding proof of McDiarmid's inequality

I am working through Wasserman's lecture notes set 2 and I am unable to fill in the missing steps in the derivation of McDiarmid's inequality (p.5). Just like my previous question in the forum, I am ...
8
votes
1answer
664 views

Bound on moment generating function

This question arises from the one asked here about a bound on moment generating functions (MGFs). Suppose $X$ is a bounded zero-mean random variable taking on values in $[-\sigma, \sigma]$ and let ...
8
votes
1answer
808 views

Understanding proof of a lemma used in Hoeffding inequality

I am studying Larry Wasserman's lecture notes on Statistics which uses Casella and Berger as its primary text. I am working through his lecture notes set 2 and got stuck in the derivation of lemma ...
4
votes
1answer
525 views

One sided Chebyshev inequality for higher moment

Is there an analogue to the higher moment Chebyshev's inequalities in the one sided case? The Chebyshev-Cantelli inequality only seem to work for the variance, whereas Chebyshevs' inequality can ...
12
votes
1answer
527 views

Probability inequalities

I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an exponential upper ...
17
votes
4answers
997 views

What is a tight lower bound on the coupon collector time?

In the classic Coupon Collector's problem, it is well known that the time $T$ necessary to complete a set of $n$ randomly-picked coupons satisfies $E[T] \sim n \ln n $,$Var(T) \sim n^2$, and $\Pr(T ...