Questions tagged [absolute-value]

Question concerns the consequences of using the absolute value function in part of the definition of one or more statistics.

From Wikipedia:

In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign.

Questions using this tag might center on:

31 questions
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Why square the difference instead of taking the absolute value in standard deviation?

In the definition of standard deviation, why do we have to square the difference from the mean to get the mean (E) and take the square root back at the end? Can't we just simply take the absolute value of the difference instead and get the expected…
saimas
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Expected value of the absolute standardized t distribution

What is the expected value of the absolute standardized t-distribution - i.e.,: $E(|X|)$ , where $X$ has the standardized t-distribution?
user261912
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Expected absolute deviation greater than standard Laplace

Could there exist a distribution, other than standard Laplace (probability density of the form $1/2e^{-|x|}$ ), on $\mathbb{R}$ such that $E[x]=0,E[|x|]=1$ and that \begin{equation*} E[|x-a|] \geq |a|+e^{-|a|} \end{equation*} for all $a \in…
Sushant Vijayan
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Is the absolute value of the difference between two Poisson distributions a Poisson distribution?

What is the distribution of the absolute value of the Skellam distribution?
Ricardo
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Percentage changes versus absolute changes when comparing rankings

We are analyzing the ranking changes of countries based on an economic index to determine whether these variations are significant. For example, Country A moves from rank 5 to rank 3, representing a 40% improvement, while Country B moves from rank…
jmatosv
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Expectation of sum of absolute values for correlated normal random variables

Let $x_1, x_2, \dots, x_{N}$ i.i.d. random variables $\sim \mathcal{N}\left(0,\sigma^2_x\right)$ . Further, let $z\sim \mathcal{N}\left(0,\sigma^2_z\right)$ , $z$ is independent from all $x_i$ . We build random variables $y_i=x_i+\gamma z$ . By…
Marius Zoican
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What is the expectation of the absolute value of the Skellam distribution?

In particular, for a Skellam distribution obtained by substracting two iid Poisson Processes. Thank you!
Marius Zoican
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Is the matrix formed by taking the absolute values of the elements of a positive-definite covariance matrix still positive definite?

Suppose I have a positive-definite covariance matrix $\boldsymbol{\Sigma}$ . I construct a new matrix $\boldsymbol{\Lambda}$ , where each element $(i,j)$ in $\boldsymbol{\Lambda}$ is equal to the absolute value of the corresponding element in…
niandra82
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Representation of the expectation of absolute value of the difference $Y-X$

Given the representations of the mean values $$E[Y]=\int(1-F(x))\,dx$$ and $$E[X]=\int(1-G(x))\,dx$$ where $F$ and $G$ are the distributions of $Y$ and $X$ respectively, can I use them to find the expression (or a bound) of $E[|Y-X|]?$ For instance,…
Jo R
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Need help understanding how only variable A can be correlated to the absolute value of A-B

I'm currently working with the dataset of a study I'm conducting. The data is comprised of serially drawn samples from patients where we've measured the cell counts of those samples and compared them to eachother to see if there's any variability…
Marcus Bådholm
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Absolute difference as disagreement measure

I am conducting a descriptive study on how the extent of disagreement in various opinions on preventive care varies depending on the specific opinion domain (i.e., prevention against different diseases) and the relationship (e.g., with parents or…
kris
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Finding the value of $k$ for an Uniform Distribution defined on $(-k,k)$

If $X$ be an uniform distribution defined on $(-k,k)$ , then the value of $k$ for so that : $$P(|X|<1) = P(|X|>2)$$ I began by defining the $p.d.f$ of the Uniform function namely: $$ f(x) = \frac{1}{2k}$$ Then we can re-write the LHS of the…
Kalvin
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Distribution of errors

I am struggeling with a basic question and would be happy to get some pointers. I am trying to evaluate an algorithm $f$ which maps some sample $x$ onto a scalar $y\in\mathbb{R}$ , i.e. $f(x) = y$ . For each each sample I have a ground-truth version…
check
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Compounding a Gaussian distribution with variance distributed according to the absolute value of another Gaussian distribution

Have there been earlier descriptions of the following compound distribution? Compounding a Gaussian distribution with variance distributed according to the absolute value or square of another Gaussian distribution: $$ f(y) = \int…
Sextus Empiricus
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Expectation of the absolute value of the product of correlated jointly gaussians?

I am reading the Performer paper https://arxiv.org/abs/2009.14794 . To understand their ReLU kernel used to approximate softmax attention, I need to evaluate $\mathbb{E}[ReLU(x^T w) \cdot ReLU(y^T w)]$ where $w \sim \mathcal{N}(0, I)$ . I was able…
N. Menet
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