Questions tagged [distributions]
A distribution is a mathematical description of probabilities or frequencies.
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On an infinite linear combination of chi-squared random variables
Question
Let $Z_i\sim\chi_{(1)}^2$ be i.i.d. chi-squared random variables with 1 degree of freedom. We define:
$$
W_{\infty} = \sum_{k = 1}^{\infty} \frac{Z_k}{2^{k}}
$$
I have interest in computing ...
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Why can a model's SHAP values change on a new dataset?
Background
I'm validating a model and as part of the process I've been calculating SHAP values for different validation datasets.
I've calculated SHAP values for every sample in each dataset taken ...
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Metrics to assess the difference between two distributions [closed]
I'd like to assess the difference between two distributions and am a bit overwhelmed by the potential amount of metrics (see result of my preliminary search below). Is there a book / review paper ...
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How to show that the influence function of minimum density power divergence estimator with positive tuning parameter is bounded?
In the linked paper, in the influence function section, the term ${u_{\theta}(y)}{f_{\theta}(y)}^\alpha$ is directly called bounded which i do not get the explanation of? Here $\alpha > 0$ is the ...
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What is the specific name of this distribution?
I just can’t seem to find the name of this distribution:
$$\frac{e^{-x}}{(1+e^{-x}) ^2}.$$
From my understanding, it is generally applied to pandemics/epidemics.
None of the statistics books that I ...
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Closed form expression for the 2-Wasserstein distance between generalized Gaussian distributions
Essentially the title - is there a closed form expression for the 2-Wasserstein distance (aka Frechet distance, Earth Mover's distance) between two generalized Gaussian distributions? The regular ...
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Example of a system with contextual statistics [closed]
What is a system described by a set of random variables for which there are distributions over subsets of these variables which are not marginal of a distribution over all random variables at once.
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Multinomial regression for modelling change of proportional makeup
I’m reaching the limits of my statistical understanding here when it comes to model specification in BRMS. I’m a PhD student researching how different drivers of species distribution (generalised into ...
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Concentration inequality for hypergeometric distribution
Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
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Can i have a distribution that is not a marginal of another distribution?
Can there be a distribution that cannot be expressed as a marginal of some other distribution?
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Does this type of distribution have a name?
I have some integer data, produced by slightly convoluted numerical procedure, which is distributed between $0$ and $300$, with the most probable values being $0$ and $300$, and the least probable ...
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Sampling from a distribution characterized by its characteristic function
Consider the following measure:
$$d\nu (x)=\mathbb 1_{(0,1)} (x) \frac 1 {x^{2}}$$
Now, define $X$ with characteristic function given by:
$$\varphi_{X}(t)= \exp\left\{ \int_{\mathbb R} [e^{itx}-1 - ...
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Posterior Predictive Distibution
How do we actually calculate (what are the operations that need to be done) the posterior predictive given a vector of observations; can we do away with the assumption of independence?
Let's say we ...
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species differential abundance across time series samples
Apologies if it is a stupid question or has been asked before or I have phrased it the wrong way, but I have been looking for an answer for days and I could not find anything - so any help will be ...
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Overlap coefficient of data distributions [closed]
For my PhD project, I am generating some artificial data, and looking at its properties. I am using ANN approaches and comparing various method's performance. The first is a VAE, which uses normal ...
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Can inverse sampling method be adapted to random vectors?
This might be a very basic question, but it seems that in all the examples I've seen, the inverse sampling method (i.e., input uniform RV into the inverse of CDF of desired PDF/probability ...
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expectation value, distribution function and the central limit theorem
The problem goes thus:
${\{X_n\}}$ is an $iid$ sequence of random variables with mean 0 and variance $\sigma^2$. If the third moment is finite, show that $$\lim_{n \to \infty} \mathbb{E} \left(\left(...
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How to identify popular items in time-series data
I have a dataset which contains time-series events in which each event is an item chosen from a unknowably large set of items. For example, let's say that the data entries are each music playlist ...
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Determine the distribution of a random variable
I want to solve the following
A salesman has two different stores where he sells computers. The
probability that he sells, in one day, a computer in the first store is $0.4$ and
independently, the ...
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What is the resultant distribution of this two-step sampling process?
This is my sampling algorithm
Let p(x) be a discrete distribution and f(x) be some function on real numbers.
Consider
$\textbf{Generate Samples from } p(x): \text{Initially, you sample } x_1, x_2, \...
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Distribution of a bootstrap sample from a given reference sample [duplicate]
Consider $n$ samples $X_1,X_2,..,X_n$, where each is normally distributed according to mean $\mu$ and variance $\sigma^2$. We use these samples to define our reference population.
Given a bootstrap ...
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Probability of Random variable less than a quantity containing random variable [closed]
What is the probability of the following: $P\left(Z_j>\frac{\epsilon}{a_j*R}\left(\sum^{M}_{m=j+1} ~ a_m*R*X_m+1\right)\right)$
where $Z_j$ and $X_m$ are independent and identically distributed ...
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For KNN: How to justify that the probability of points to fall into a sphere of volume $V$ is $p(X)V$
https://www.cs.cmu.edu/~lwehbe/10701_S19/files/Lecture_3.pdf
At the end of these notes, there is a short paragraph.
Let $x$ be a test point. Let $x_1, \ldots, x_K$ be its $K$ nearest
neighbors. Let $...
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What's the best clustering algorithm for Fraud Data?
Background
I'm working on a Fraud dataset with 500,000 samples, and 130 features.
There are:
98 numerical features,
32 categorical features,
There are missing values in:
7 numerical features,
12 ...
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answers
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Approximating the distribution of the product of iid beta variates
Background
I am interested in the distribution of
$$\theta_0=1-\prod_{i=1}^n(1-\theta_i)$$
where the $\theta_{i>0}$ are iid beta random variates:
$$\theta_{i>0}\sim\text{Beta}(\alpha,\beta)$$
In ...
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Comparing the output distribution of two ML models
Consider a regression task (e.g. predicting house prices) with a given train and test sets.
We start with constructing a linear regression model, in which we assume $y_i=X^T\beta+\epsilon$ with $E[\...
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Calculating the cumulative distribution function and the probability density function of an interval with ratio of a shorter and longer segment
The interval $[0, 2]$ is divided into two parts by randomly marking a point in $[0, 1]$ according to the rectangular distribution. Let $X$ be the length ratio $L_1/L_2$ of the shorter segment $L_1$ to ...
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In a 2-arm clinical trial where ten centers recruited patients, how does one test that two distributions are statistically similar?
There are two arms in a clinical trial. If ten centers recruited subjects. How do I test in R that distribution across centers into two arms are statistically same? The distribution appears ...
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Methods for fitting a distribution to regression data
I'm hoping to find a method/algorithm/approach for fitting to a distribution to regression data.
Essentially I have a problem where I have survival data with independent variables, but only cases that ...
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1
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How to approach problem [closed]
I have a discrete random variable f(phi) with known probabilities for each n outcomes. Phi represents the random variables m parameters. Each parameter is its own independent discrete random variable ...
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GLM: invalid value encountered in log special.gammaln
I've never used GLM before so I would like to have some hints on how to use it and if I'm missing any steps.
My challenge:
I want to know if the price of product is influenced, positively or ...
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Are there any research papers which show why Wasserstein distance is better than Jensen-Shannon/KL_div/Bhattacharya distance for specific use cases?
I am trying to find reliable research work which show why displacement based metrics such as Wasserstein distance is a better suited metric than Jensen-Shannon distance in specific use cases and for ...
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Estimating distribution function in a repeated measurement design
Suppose we have a repeated measurement design with imbalanced clusters so that $X_{1,1}, X_{1, 2}, \ldots X_{1, n_1} \sim F_1$, $X_{2,1}, X_{2,2}, \ldots, X_{2,n_2} \sim F_2$, and $ X_{m, 1}\ldots, ...
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How to measure the difference between two distributions of the same family?
Kullback-Leibler divergence seems to be a frequently used "metric" to measure the difference between probability distributions, regardless of their respective families. However, I would like ...
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Representation by Poisson transform
In physics papers that dealing with multiplicity distributions in high energy collisions, I have met with that some probability distributions can be expressed as a Poisson transform. See, eg., ...
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Hypothesis Testing - Varying p-value issue in my MWU
I have two machines (A & B) in our production line. A is older, B is very new. Both perform the same tasks, except that machine B is advanced and will perform 1 task less than the no of tasks in ...
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Is this the formula for the conditional covariance of normally distributed random variables? [duplicate]
Assuming $X$, $Y$, and $Z$ are normally distributed random variables, is it true that:
$Cov[X, Y | Z] = Cov[X, Y] - Cov[X, Z]Cov[Y, Z] / Var[Z]$
Could you provide a simple derivation?
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Choosing a probability distribution for 4D data: dirichlet challenges and alternatives
I'm seeking the right distribution for my 4D data, where the sum of values in each sample equals one. Currently, I've chosen to employ the Dirichlet distribution. However, upon applying this ...
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Pinsker-type inequality for moments?
Let $f_1$, $f_2$ be two discrete probability distributions. By Pinsker's inequality, the Kullback-Leibler divergence $D(f_1||f_2)$ sets an upper bound on the total variation distance between the two ...
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Goodness-of-fit test for very skewed data [closed]
I am working with two quite large datasets (9000 obs and 3800 obs). Each observation has been grouped into 1 of 12 categories and I am looking to test if the frequency of categories in the smaller ...
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Efficiently sample from a limit set given a differential equation?
Given a dynamical system of many variables, described by an ordinary differential equation, is there some way to use machine learning to efficiently sample from the limit set (or maybe more accurately ...
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Random Variable with E and V
Question is Let X be a random variable with $E(X)=1$ , $V(X)=5$.
I need to find (a) $E[(1+X)^2]$ and (b) $V(4+3X)$.
I think I solved (b) alright. Please let me know if there is an error.
For (b), I ...
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What is the effect of treating the samples of a non stationary process as stationary and plotting their distribution
Say I have a non stationary process. Let's say I take one realization of this process given by finite number of random variables $X_1$, $X_2$,...$X_n$. Now I plot them on a histogram. Can this ...
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Is it possible to make a Poisson Distributions without Poisson Property?
In Poisson Distribution, the amount of time you need to wait for next event does not depend on how much time you have already waited. EX: In Poisson Distribution with parameter lambda, event happens ...
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On the estimated formula of covariance of two random variables
We define the covariance of two random variables $X$ and $Y$ as $Cov(X,Y) = E[(X - \mu_X)(Y - \mu_Y)]$. The covariance measures the "linear dependence" between the two r.v s.
But in a lot of ...
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Bayes's theorem problem issues
Problem Statement:
Assume that 40% of all interstate highway accidents involve excessive speed by at least one of the drivers (event $E$) and that 30% involve alcohol use by at least one driver (event ...
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Fitting data with a beta distribution using fitdist from fitdistrplus package in R
I have a variable x that corresponds to data from a database. I've been trying to find the best distribution to fit it and looking at the histogram I figured either a normal, weibull or logistic ...
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distinguishing gradients of samples in semi supervised learning under mixup
I'm having trouble drawing conclusions from a specific semi supervised machine learning setting. My use case would be using mixup [Zhang2017] and label guessing similar to the mixmatch paper [...
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Problem with Cumulative distribution function
I can't understand this cumulative distribution function. I would like to calculate the data distribution function:
...
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Likelihood Ratio Testing for Binomial Distributions
I have a feeling this is a silly question. I am working on a research paper, at some point in it we perform a likelihood ratio test. The first guess would be to apply Wilks's theorem. However, if we ...
