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Questions tagged [probability-inequalities]

Probability Inequalities are useful for bounding quantities that might otherwise be hard to compute. A related concept is a concentration inequality, which specifically provides bounds on how far a random variable deviates from some value.

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Usefulness and validity of Alternative definitions of “quantile”

According to textbook, the $p\,$th quantile of a random variable $X$ is any real value $x$ satisfying $P(X \geq x)\geq 1-p$ and $P(X \leq x) \geq p$. Why isn't the alternative definition, a $p\,$th ...
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A consistent estimator with infinite expectation?

Typical (or common) approaches to prove an estimator is consistent require finite mean and variance. The proofs usually follow from concentration bounds, e.g. Markov, Chebyshev, etc. I'm wondering ...
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Probability of sparse spectrum

Consider a vector $v$ such that $v \sim \mathrm{Unif}(\mathbb{S}^{d-1})$, the uniform distribution on the unit sphere in $d$ dimensions. Question: is there an upper bound on the probability that $v$ ...
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Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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Unusual Markov inequality for normal distribution

I'm trying to answer the following question from Larry Wassermans book on statistical inference. My question is how did they arrive at the Markov bound, it does not seem like the normal form of the ...
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Getting From Concentration Inequality to Interval Length

I've seen this used some times and I would like to ask what steps are taken on the way to getting there: E.g. assuming bounded variance, we can use Chebyshev concentration inequality: for any $t>0$...
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General solution of expected value of E(f(X))?

This is maybe a trivial question I came up while solving a few examples and understanding Markov/Chebyshev inequalities and subsequently in evaluating Chernoff bounds. Suppose $X$ is a random variable ...
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Best confidence interval for sample mean of any variable in L2

I was thinking about a very basic matter, and arrived at the conclusion I know a lot less about it than I thought. When people have data $X_1^N$ sampled i.i.d. from a distribution: $X_1^N \sim X$,...
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Bounds on quantiles of the minimum of summations of (possibly dependent) random variables

Suppose I have $2N$ continuous random variables $X_1, \ldots, X_N, Y_1, \ldots, Y_N$ and that I can evaluate the quantiles of the respective distributions. Given a value $w \in [0, 1]$ I would like to ...
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Does an inequality hold as an expectation over a probability distribution?

Suppose I have to functions $f(x)$ and $g(x)$ such that $$f(x) \leq g(x) \quad \forall x.$$ For a distribution $\pi(x)$ on $x$, is it necessarily true that $$E_\pi[f(x)] \leq E_\pi[g(x)]?$$ My ...
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Simplification using Cauchy Schwartz Inequality

Can someone please help me understand how the last step in the highlighted part in the equation is arrived at? I did not get how the Cauchy Schwartz Inequality comes into play here. The full paper is ...
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Intuition behind the no convergence of the variance of sum of random variables

$$Var[\bar{X}] = \sigma^2/n$$ $$Var [\sum{X}_i] = n\sigma^2$$ $$lim_{n \to \infty} Var[\bar{X}] = 0$$ wich means at $\infty$ we will always get the same $\bar{X}$ after every simulation. I ...
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Effect of scaling on the mean of random variables

Consider two possibly correlated scalar random variables $N$ and $X$. It is known that $1\leq N \leq N_{\max}$. Given that $\mathbb{E}[NX] \leq 0$, does it always hold that $\mathbb{E}[X] \leq 0$? ...
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How to prove the concentration equality for standard normal?

The following inequality is given in some of Yale's online lecture notes $$P(|Z|>x) \leq 2 \sqrt{2 \pi} \phi(x)$$ Where $Z \sim N(0,1)$ with density $\phi(x)$. They call it a concentration ...
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Tight upper bound on the expectation of a concave function

N is a random variable whose sample space is [0,$\infty$). I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The ...
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