Questions tagged [symmetry]
The symmetry tag has no usage guidance.
58
questions
1
vote
1answer
13 views
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: ...
2
votes
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 ...
0
votes
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 ...
0
votes
0answers
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 $\mu_1^j$ ...
0
votes
0answers
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 ...
0
votes
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
$$
\...
0
votes
1answer
32 views
Definition of symmetric probability mass function [duplicate]
Consider the random variables $Y$ characterised by a probability mass function (PMF) as follows:
$$
Y=\begin{cases}
-2 & \text{ with probability $\frac{1}{2}$}\\
2 & \text{ with probability $\...
0
votes
0answers
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 (\...
1
vote
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 (\...
2
votes
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 ...
4
votes
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 ...
5
votes
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 ...
1
vote
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 ...
3
votes
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 $(X,Y,Z)$ ...
2
votes
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
votes
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| ...
2
votes
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 ...
0
votes
0answers
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 ...
1
vote
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_{...
2
votes
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 ...
0
votes
0answers
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}$,...
1
vote
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 ...
2
votes
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?
3
votes
0answers
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 ...
1
vote
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
votes
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 ...
6
votes
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 ...
1
vote
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.
1
vote
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:
...
0
votes
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}{...
1
vote
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 ...
0
votes
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 ...
0
votes
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
$$
P ( X \le a - x ) = P ( X \ge a + x ) \qquad \forall x \in \mathbb{R}
$$...
7
votes
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 ...
2
votes
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=...
1
vote
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.
...
0
votes
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, ...
1
vote
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?
5
votes
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 ...
30
votes
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
votes
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'...
1
vote
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 ...
3
votes
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?
4
votes
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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
votes
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 ...
1
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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 ...
4
votes
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?
