# Questions tagged [distribution-tails]

This tag is for questions relating to "tail-distribution" which essentially means how much probability is distributed over the largest values(usually) of the random variable.

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### Upper bound for a squared zero-mean sub-Gaussian random variable: Transforming a sub-Gaussian rv to a a Gaussian rv

I'm studying from the book "Mathematical Analysis of Machine Learning Algorithms" by Tong Zhang. Theorem 2.9 states Let $\{X_n\}_{n=1}^N$ be independent zero-mean sub-Gaussian random ...
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### Approximating a modified Gaussian integral by considering tail-end behavior

I am considering calculating the following definite integral. The limits of integration are positive and $c > 0$. One can see that the integrand is bounded above by the Gaussian in the numerator ...
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### Are these two versions of the Hanson-Wright Inequality Equivalent?

I have a question regarding the following two different versions of the Hanson-Wright inequality for estimating the tail behavior of (sub-)Gaussian chaos (i.e. quadratic forms). The first is Thm. 6.2....
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### $X$ sub-Gaussian $\implies \text{Var}(X) \leq \sigma^2$

$\newcommand{\V}{\text{Var}}$ $\newcommand{\E}{\mathbb E}$ Definition: A mean zero random variable $X$ is $\sigma$ sub-Gaussian if for all $\lambda \in \mathbb R$, \begin{align} \E\left[\exp\left(\...
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### The lower bound probability of Bernoulli sum is positive [closed]

Let $X_i$ be iid random variables. It equals to $1$ with probability $(1+p)/2$ and equals to $-1$ with probability $(1-p)/2$. My question is: What is the probability that $\sum_{i=1}^nX_i>0$? I ...
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### Asymptotic expansion for the density-quantile function of a bijected truncated normal distribution

Summary How would I prove that $$fQ(u) = \frac{1}{a} \left( 1 - \Phi^{-1}(au+b)^2 \right)\varphi\left(\Phi^{-1}(au+b)\right) \sim (1 -u),$$ as $u \to 1$, where $\Phi, \varphi$ are the CDF and PDF of ...
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### tail estimation of normal distribution

In https://www.johndcook.com/blog/2021/11/05/normal-tail-estimate/ , why "base of our rectangle runs from x to the point t where the integrand drops by a half" ?
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