Questions tagged [symmetry]

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Is there a signed (ie anti-symmetric) version of SMAPE?

The symmetric mean absolute percent error (SMAPE) is a symmetrized version of percent error with the formula: $$\frac{200\%}{n}\sum_i\frac{|x_i - y_i|}{|x_i| + |y_i|}$$ SMAPE is symmetric: ...
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2answers
80 views

What is a symmetric distribution symmetric about if it has zero skewness? [duplicate]

We know that a distribution with zero Skewness are symmetric. A quick Google search or looking up in textbooks says that Symmetric distributions are distributions where the left side mirrors the ...
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1answer
60 views

What would be the output distribution of ReLu(X) activation (In case that the distribution of X is unknown)?

Suppose E[X]=0, var(X)=1 and we know X has a symmetric distribution, What would be the distribution of 𝑌=ReLU(𝑋)=max{0,𝑋}? I have seen this question What would be the output distribution of ReLu ...
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16 views

Wassertein “least squares” and symmetries

So here's a scenario: I have points $(\mu_1^j,\mu_2^j)$ and I associated them the following distribution $$\rho_j=1/2\delta_{\mu_1^j}+1/2\delta_{\mu_2^j}$$ These have symmetry (exchanging $\mu_1^j$ ...
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8 views

Symmetry vs asymmetry [duplicate]

If a collecƟon of numerical values X is roughly symmetric the value for the mean and median are about the same. However, if the mean and median are about the same value this does not necessarily mean ...
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0answers
23 views

Showing that if the PMF of $W$ is symmetric around zero then some parameters entering it are equivalent

Summary: In what follows, I specify the probability mass function (PMF) of a random variable $W$, depending on some parameters $(\lambda,\mu,\lambda',\mu')$. I would like your help to show that $$ \...
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1answer
32 views

Definition of symmetric probability mass function [duplicate]

Consider the random variables $Y$ characterised by a probability mass function (PMF) as follows: $$ Y=\begin{cases} -2 & \text{ with probability $\frac{1}{2}$}\\ 2 & \text{ with probability $\...
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23 views

Show that the intersection of two sets involving symmetric PMF is empty

Consider the stepwise cumulative distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $J<\infty$ $\lambda\equiv (\...
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1answer
25 views

Characterise the set of symmetric probability mass functions

Consider the stepwise cumulative distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $J<\infty$ $\lambda\equiv (\...
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1answer
82 views

Symmetry group in posterior distribution/inference

Here's a scenario: Suppose I collect a dataset $\{x_i\}_{i=1}^k\subseteq\mathbb R$ of data points $x_i$, and I wish to explain it using a mixture of two Gaussians; assume the unknown parameters are ...
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1answer
111 views

What can we say about distributions of random variables $X$ such that $X$ and its inverse $1/X$ have the same distribution?

What can we say about random variables such that it and its inverse have the same distribution? One example is Cauchy distributed random variables, easily proved via the fact that if $X, Y$ are IID ...
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2answers
184 views

Identically distributed vs P(X > Y) = P(Y > X)

I've two related propositions which seem correct intuitively, but I struggle to prove them properly. Question 1 Prove or disprove: If $X$ and $Y$ are independent and have identical marginal ...
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1answer
55 views

When $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim(X_0-X_1, X_0-X_2)$?

Consider a bivariate probability distribution $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question: Are there necessary and sufficient conditions on the CDF associated with $P$ (joint or ...
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0answers
85 views

From bivariate to trivariate probability distribution

Let $\mathcal{G}$ be the space of all possible bivariate probability distributions. Let's pick a bivariate probability distribution $g\in \mathcal{G}$. Can we always find a random vector $(X,Y,Z)$ ...
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1answer
38 views

How to specify uniform distribution with same properties as normal distribution?

What I mean is, is it possible to specify a uniform random variable $U$ with random parameters $a,b$, where $a=-b$, and are generated from some other distribution, such that the marginal pdf of $U(a,b)...
4
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1answer
44 views

A symmetric iid process

Let $X_1, X_2, \ldots$ be an iid process with $X_i$ having a symmetric distribution around $0$. Then can I always write $$X_1 - \alpha X_{t-1}-\alpha^2 X_{t-2}-\cdots \stackrel{iid}{=} X_1 + |\alpha| ...
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1answer
39 views

It what situation is a distribution known to be symmetric, but about an unknown location?

A favorite example in theoretical statistics is this: A sample of individuals are drawn independently from a distribution with density $f(x)$, where $f(x)$ is unknown, but is known to be symmetric ...
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14 views

symmetric marginal but asymmetric joint distribution contours [duplicate]

Let us say we have two continuous random variables, $X$ and $Y$ such that their pdfs $f(x)= f(-x)$ and $g(y)= g(-y)$ for all $x$ and $y$. In other words, $X$ and $Y$ have symmetric distributions ...
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1answer
25 views

How to test for the symmetry of a finite sequence?

I have a finite sequence of real numbers ${\{a_n\}}_{n=0}^{N-1}$, for the sake of simplicity I assume $N\gt1$ is even. The sequence is symmetric (I would say even like an even function) iff $a_k=a_{...
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0answers
76 views

Testing for symmetric distributions

Suppose we have $n$ samples $s_1,...,s_n$ from an unknown real-valued distribution $D$. We are interested in a statistic to test if $D$ is symmetric around zero. (In my application, $n$ is only ...
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33 views

Categorical probability distribution that captures “some” permutation invariance / mirror symmetry

I'm fitting something similar to a naive Bayes model to a data set where each data point has six features, $A_1$, $B_1$, $C_1$, $A_2$, $B_2$ and $C_2$. $A_1$ and $A_2$ can both take values in {$a_{1}$,...
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1answer
34 views

How can I cluster data drawn from distributions with known symmetries?

Consider a set of data which is a mixture of samples drawn from different distributions. It is known from the underlying phenomena generating the mixture that for every distribution in the mixture ...
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1answer
405 views

Z distribution is symmetric. Chi square distribution is not symmetric. Why?

Z distribution is symmetric. Chi square distribution is not symmetric. Why?
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40 views

Comparison between variance of $|x|$ and $x$ for the symmetric distribution

For a symmetric distribution, how the following inequality holds which is given by my teacher: $V(|X|)>V(X)$ What I think is that it should be opposite since for a symmetric distribution the mean ...
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0answers
18 views

The degree of nonparametric estimation kernels and induced $U$-statistics

The definition of kernels in nonparametric can be formulated as follows. [Randles&Wolfe] pp.61-62. A parameter $\gamma$ is said to be estimable of degree $r$ for the family of distributions $\...
4
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1answer
186 views

linear regression, symmetry of model does not lead to symmetry of coefficients

Experiment: You are given a large population of real numbers. For simplicity take the whole numbers from -n to n. Take two independent random samples x and y of size k and sort them (each one ...
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2answers
353 views

$X$, $Y$ independent identically distributed. Are there counterexamples to symmetry of $X-Y$?

That $X-Y$ should be symmetrically distributed for iid $X,Y$ is obvious simply by interchanging the roles of $X$ and $Y$ -- informally we might argue Let $Z=X-Y$ have distribution $F$. The roles of ...
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3answers
210 views

Symmetry of a ratio of two random variables

My question is whether the fact that two random variables, X and Y, are symmetrically distributed implies that their ratio, Z=X/Y, is symmetrically distributed too.
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1answer
21 views

Question related to symmetry in distances

My dataset represents products and evaluation of every product by users. E.g., I might have: ...
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2answers
45 views

Identity related to symmetric distribution

Let $F$ be a symmetric (around $\frac{1}{2}$) cumulative distribution whose support is $[0,1]$. So, $F(x)=1-F(1-x),\forall x\in[0,1]$. Would this identity hold for any such $F$? $$\int^1_0(x-\frac{1}{...
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0answers
167 views

How to calculate growth rates up and down from a local maxima?

Problem: I am currently doing my Msc-thesis on rodent population dynamics. One of my aims is look at symmetry in oscillation topography. For this I want to calculate the growth rates up and down from ...
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0answers
28 views

regarding symmetric distributions [duplicate]

i asked a similar question yesterday , i think my way was not proper ,i received vague and confusing answers, so i asked it again clearly specifying what i actually want to ask This is what i found ...
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2answers
4k views

Definition of symmetric random variable [in terms of distribution function ] [duplicate]

This is what i know about symmetric distributions: The distribution of rv (random variable) $X$ is symmetric about $a$ iff $$ P ( X \le a - x ) = P ( X \ge a + x ) \qquad \forall x \in \mathbb{R} $$...
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2answers
339 views

Prove that a distribution is symmetric using moments

Given, a random variable X whose mean , variance and fourth central moment are 0, 2 and 4 respectively. Now, how do I prove that (1) third moment is 0 (2) distribute is symmetric about 0 and (3) X ...
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0answers
51 views

Writing monomial symmetric functions in terms of population moments about the mean

Following Sukhatme (1954, pp.35 - 36) for a univariate case, I encountered the following population monomial 'symmetric' functions for a bivariate case: $\sum_{i=1}^{N}(X_i^2Y_i^2)$ $;$ $\sum_{i\neq j=...
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0answers
62 views

About symmetric distributions

Homework problem: if $X$ and $Y$ are two independent random variables, where $Y$ is symmetric about 0, define $U = X + Y , V = X- Y$. Then say whether $U$ and $V$ have the same distribution or not. ...
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1answer
52 views

Generate multivariate symmetric distribution

I have a (bivariate) spherically symmetric distribution, in the sense that I can generate iid values distributed according to it. But there is a detail - I can't get the generated values as a whole, ...
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1answer
2k views

How to prove conditional independence symmetry $X\perp Y | Z$? [closed]

In probabilistic graphical modeling, conditional independence $X\perp Y | Z$ means $P(X,Y|Z)=P(X|Z)P(Y|Z)$. How to prove it?
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1answer
3k views

Why are mean and median not equal for asymmetric distributions?

My reasoning is as follows: the p.d.f. is divided by the mean (expected value) into two parts, for which the areas under the p.d.f. curve are equal, hence the probabilities that random variable takes ...
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4answers
6k views

Does mean=mode imply a symmetric distribution?

I know this question has been asked with the case mean=median, but I did not find anything related to mean=mode. If the mode equals the mean, can I always conclude this is a symmetric distribution? ...
4
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1answer
582 views

How to prove the test statistic of the Wilcoxon signed rank test is symmetric about mean

How to prove the test statistic of the Wilcoxon signed rank test is symmetric about its mean? I know that if I want to prove a distribution is symmetric, I need to show $f(m-t)= f(m+t)$ but $W$ doesn'...
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0answers
72 views

Symmetry in supervised learning models

I am training a bunch of supervised models on a binary classification problem. My dataset is comprised of some positive examples (p) and their symmetric negative examples (n), i.e. rows created by ...
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1answer
580 views

How to understand boxplot skewness

I have a boxplot that is drawn and the whiskers are the same length but the median is closer to the upper quartile than the middle. Would this be considered skewed or symmetrical?
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0answers
139 views

How can I test whether an empirical discrete distribution is symmetric?

For example, I have a distribution with mean = 55.46; med = 54.5; mode = 45. The Shapiro-Wilk is non significant, the data are unimodal, and there is no significant skew or kurtosis. At a glance, the ...
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0answers
73 views

A More Symmetric Distribution?

REVISED QUESTION I am interested in learning more about why certain confidence intervals are better than others in a logistic regression setting. Based on some readings and discussions with more ...
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2answers
161 views

Quantify the asymmetry of a one dimensional vector

I have image data, which I can represent as one dimensional vectors. Each value represents the brightness of a pixel in a line. eg: (1, 12, 4, 3, 1, 4) I want ...
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0answers
54 views

Test symmetry of effects of a categorical variable in linear model?

I have a linear model in which x and a categorical variable explain Y. The categorical variable contains 3 categories a positive, a negative and a neutral category. I am using the neutral category as ...
2
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1answer
291 views

Is there a Bowker-like test to assess the symmetry of squared contingency tables with small expected values and a small number of observations?

I ask this question because Bowker test (McNemar extension for $k\times k$ tables with $k>2$) requires a large number of observations for fitting well and also requires (as a rule of thumb) that ...
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0answers
71 views

Distribution of correlation coefficient in compound symmetric covariance model

Suppose $X_i \overset{iid}{\sim} N_m(0, \Sigma)$ for $i=1, \dots, n$ for the case that $$\Sigma = \sigma^2((1-\rho)I_m + \rho \mathcal{1}\mathcal{1}^\prime),$$ where $\mathcal{1} \in \mathbb{R}^m$ is ...
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2answers
2k views

a perfect or symmetric bimodal distribution

I would like to know how I can measure the degree of symmetry of a bimodal distribution. Is there any a criterion like for example skewness in the case of unimodal distributions?