# Questions tagged [symmetry]

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### Which test(s) are alternative to Mann-Whitney test for non-parametric continues data when symmetry assumption is violated

From here and here I see that we cannot use Mann-Whitney test if symmetry assumption is violated. Which test(s) can we use instead of Mann-Whitney test for non-parametric continues data if symmetry ...
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### Asymmetric robust regression

What are the methods for robust regression with asymmetric distribution of outliers? I am specifically interested in equivalents of Huber and Tukey M-estimators. However, asymmetric heavy-tailed ...
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### 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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### 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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### 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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### 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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### 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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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 $U(a,b)... 1answer 50 views ### A symmetric iid process Let$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| ... 1answer 42 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 ... 0answers 19 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 ... 1answer 33 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_{... 0answers 108 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 ... 0answers 35 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},... 1answer 40 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 ... 1answer 614 views ### Z distribution is symmetric. Chi square distribution is not symmetric. Why? Z distribution is symmetric. Chi square distribution is not symmetric. Why? 0answers 49 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 ... 0answers 29 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 \... 0answers 31 views ### Symmetry of circular distribution I wanted to ask somebody for an opinion about one thing I found in the statistical book (Mardia, Jupp - Directional statistic). They were defining circular symmetric distribution in following sense: ... 1answer 239 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 ... 2answers 509 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 ... 3answers 287 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. 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: ... 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}{... 1answer 25 views ### Would there exist a symmetry around the mode in a truncated uni-modal distribution (which is differentiable)? If we truncated around the mode of an asymmetric (continuous and differentiable) unimodal distribution, would there be a symmetry around the truncation point? For example if X is generated from an ... 0answers 185 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 ... 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 ... 2answers 5k 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 $$P ( X \le a - x ) = P ( X \ge a + x ) \qquad \forall x \in \mathbb{R}$$... 2answers 479 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 ... 0answers 52 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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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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### 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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### 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? EDIT: this post might help your understanding https://math.stackexchange....
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### 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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### 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? ...
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### 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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### 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 ...