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Questions tagged [absolute-value]

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

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Distribution of a spread of observations in triplicate sample taken from Gaussian distribution

Suppose random triplicate samples are taken from a Gaussian distribution with known mean and SD. What should be the distribution of the maximum absolute difference between 3 possible pairs of ...
Maciej Tomczak's user avatar
1 vote
0 answers
76 views

Estimating variance of a Gaussian distribution from the mean of absolute values of first differences in a sequential sample

Edited Question: This question is about estimating the variance (or $\text{SD}_0$) of a parent distribution from which a set of random independent samples is taken. The samples are taken in sequence ...
Maciej Tomczak's user avatar
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If A and B are correlated, what is the correlation between |A1 + A2| and |B1 + B2|?

To preface, let random variables $X = A_1 + A_2$ and $Y = B_1 + B_2$. $A_1$ and $A_2$ are copies of $A \sim N(0, 1)$, and $B_1$ and $B_2$ are copies of $B \sim N(0, 1)$. In this situation, if $A$ and $...
Vance's user avatar
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7 votes
1 answer
208 views

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|+...
Sushant Vijayan's user avatar
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0 answers
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Probability that a random variable has a greater absolute value than the sample mean of iid random variable

I have encountered a problem which involves finding the probability that an observation of a random variable has a greater absolute value than the sample mean of an independent set of observations of ...
chasmani's user avatar
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5 votes
1 answer
299 views

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 ...
Jo R's user avatar
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2 votes
1 answer
195 views

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 $\mathbb{E}[ReLU(x^T w) \cdot ReLU(y^T w)]$ ...
N. Menet's user avatar
2 votes
1 answer
146 views

Convergence in probability to a constant and absolute value (?)

I am a bit loss with the convergence in probability and the absolute value. Let $X_n$ be a random variable defined in $\mathbb{R}$ with $\lim_{n \rightarrow \infty} E[X_n] = a$ and $V[X_n] = O(n^{-1})$...
Eryna's user avatar
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Sampling distribution of the slope estimator of a stationary non-linear AR(1)

Consider a stationary non-linear AR(1) model $$x_t = |\theta x_{t-1}|+w_t,$$ where $w_t$ is i.i.d. standard normal. Given a sample $\{x_0, \cdots, x_n\}$, I want to find an estimate of $\theta$ and ...
The One's user avatar
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1 answer
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Solve an inequality finding the upper bound

Suppose that there exists a constant $C$ such that the following relation holds for all $G$: \begin{equation*} \vert T(F)-T(G) \vert \le C \sup_y \vert F(y)-G(y) \vert \end{equation*} Suppose that ...
Bibi's user avatar
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66 views

Test which time series is closer to zero

I have two time series, A and B, which represent reactions of subject A and B to a certain medication measured at different point in times. I want to formally test that the reaction of patient A is ...
user9875321__'s user avatar
0 votes
1 answer
263 views

MGF of the absolute Value of a Skellam RV

I am trying to derive the moment generating function for the absolute value of a Skellam random variable $Skellam(\lambda_1, \lambda_2)$ Suppose $X_1 \sim Pois(\lambda_1)$ and $X_2 \sim Pois(\lambda_2)...
Lewkrr's user avatar
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4 votes
1 answer
187 views

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) = \...
Kalvin's user avatar
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Expectation of absolute random variable and its relationship with absolute expectation value

For any continuous random variable $X$, it is obvious that $|E X| \leq E|X|$. My question is, what kind of distribution $P$, such that $X\sim P$ and satisfy $|E X| \geq c E|X|$ for some positive $c\in ...
DoubleL's user avatar
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4 votes
3 answers
176 views

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 ...
check's user avatar
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5 votes
2 answers
139 views

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 ...
Marcus Bådholm's user avatar
2 votes
0 answers
81 views

An interesting non-smooth regression

Consider the following setup: Let $x_1^k$ and $x_2^k$ be length $N$ vectors of observed reals, where $N$ is about, say, 100,000, $k\in 1:K$, and $K$ is about, say, 200. (So $x_1,\;x_2$ can be thought ...
Matterhorn's user avatar
6 votes
1 answer
1k views

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 ...
Marius Zoican's user avatar
8 votes
1 answer
1k views

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's user avatar
4 votes
1 answer
617 views

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 ...
Sextus Empiricus's user avatar
1 vote
0 answers
33 views

Why is there a need to find a variance and take the square root to get a standard deviation? [duplicate]

My question is why is the formula for finding the standard deviation of a given data (either grouped or non grouped) the way it is? so let me start from the definition of a standard deviation with my ...
EHM's user avatar
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6 votes
1 answer
1k views

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's user avatar
2 votes
3 answers
3k views

Why should I prefer the standard deviation over other measures of variance? [duplicate]

The most common kind of deviation is the standard deviation. $$ \text{Sd}(x) = \sqrt{\text{Mean}((x - \text{Mean}(x))^2)}$$ The standard deviation is very similar to the mean absolute deviance or $$...
Ms. Molly Stewart-Gallus's user avatar
6 votes
2 answers
2k views

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's user avatar
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562 votes
23 answers
326k views

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 ...
c4il's user avatar
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