Questions tagged [distributions]
A distribution is a mathematical description of probabilities or frequencies.
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questions
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Applications of Markov, Chebyshev, Chernoff and Hoeffding inequalities
Can someone help me. I want some examples of applications of the Markov, Chebyshev, Chernoff and Hoeffding inequalities in known distributions such as Uniform, Exponential and Normal distributions.
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A question about fisher's z transformation detail of his paper published in 1921
Typically, people will use Fisher's z-transformation (arctan) to turn the r into a variable that is approximately normally distributed. When I look though his paper published in 1915(https://www.jstor....
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1answer
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Comparisons of distributions: How to deal with discrepancies between visual impression and statistical tests (K-S, ranksum)?
I'm having some doubts how to properly compare my data (because of visual discrepancies), represented as histograms.
Here's an minimal and representative example, illustrating the data:
I measure BMI ...
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2answers
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How to match the distribution of one dataset when subsampling from another?
I have two datasets that measure the same variable under different conditions. The first dataset (d1) is much larger (9,000 observations) than the second dataset (d2; 900 observations).
A histogram of ...
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2answers
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Fitting a distribution like exponential but with always negative $\frac{d^2}{dx^2} \log \text{pdf}(x)$
I have some continuous data in the domain $[0,\infty]$ which I have physical reason to believe is almost, but not quite exponentially distributed. The difference between my idea of a distribution and ...
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1answer
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Convergence Distribution and Probability [closed]
Suppose that $|X_n - Y_n|$ converges in probability to 0, and that $X_n$ converges in distribution to X. Show that $Y_n$ converges in distribution to X.
Thanks in advance.
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1answer
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Deriving distribution for multiplayer game results from pairwise probabilities
Suppose there is a game with three participants: Player A, Player B, and Player C. One player will finish in first place, another in second place, and another in third place (no ties allowed). I know ...
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Do Particle Filters actually approximate the posterior distribution?
Im reading a tutorial paper about particle filters (Link) in which it is stated that as the number of samples tends to infinity the approximated posterior density given by
$p(x_k|z_{1:k}) \approx \...
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How to sample from different datasets such that they have similar distributions?
I have data from multiple datasets with the boxplot given below
In the above figure, I have data from 7 different datasets. I am looking for a sampling strategy without replacement such that samples ...
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1answer
34 views
What is the MLE of the Continuous Bernoulli distribution?
The continuous Bernoulli is a distribution I recently discovered. What the maximum likelihood estimate of the distribution's parameter? I'm struggling with the normalizing constant.
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Gaussian distribution with Kronecker product in the Covariance matrix
Assume we have two correlated multivariate Gaussian random variables $\mathbf{d_1}$ and $\mathbf{d_2}$ both distributed as $\mathcal{N}(\mathbf{0},\mathbf{R})$. We also know that $\mathbf{d_1}-\mathbf{...
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Generate data that matches a frequency distribution while preserving the original spatial structure
I am dealing with a 3D array containing values representing the "importance" of each voxel. For my analysis, I would like to synthesize n new arrays from my original array to have a ...
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1answer
72 views
R Confidence Intervals for quantiles from Generalized Lambda Distribution
I'd like to compute confidence intervals in R for quantiles from generalized lambda distribution.
Steve Su (2009) introduces below 2 ways to calculate confidence intervals. I think I could understand ...
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1answer
84 views
Expected value of (continous) exponential distribution proof/derivation
I started with the following exponential distribution:
$$ f_{exp}(x;\lambda) = \lambda\, e^{-x\lambda} \quad \forall\, x \in \mathbb{R}^+ $$
I know from internal courseslides and wikipedia that the ...
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23 views
Hypothesis testing — time between two actions
I am a complete newbie to statistics, and have gotten stuck on a difficult real-world problem.
What I'd like to do is demonstrate confidence that, for a set of < 50 observations of two actions, the ...
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What is the distribution of the time-to-ruin for a gambler's ruin problem that allows “pauper bets”?
In another question on this site I have derived the distribution for the time-to-ruin in the gambler's ruin problem where the wealth of the gambler follows a discrete-time random walk. In this ...
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Finding the (approximate) beginning of a novel based on the distribution of newline characters
I am working on a NLP project in the domain of literature. My dataset consists of a collection of books but most of the books are prepended with meta-information such as the copyright, the outline, ...
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Prove that the conditional distribution of a normal random variable is also normal random [duplicate]
How to prove the claim that the conditional distribution of a normal random variable is also normal random? And how to think it intuitively?
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Are sample means ordered by strict second-order stochastic dominance throughout the support?
Consider random variables $X_1,X_2,\dots$.
Each $X_i$ is independent and identically distributed on $[0,1]$ with a cumulative distribution $F$ that has a positive density $f(x)>0$ throughout the ...
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1answer
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Use of Change of Variables (in probability distributions) in Machine Learning
I am learning about machine learning from a probabilistic perspective via Kevin Murphy's so far fantastic Textbook (2021) Machine Learning - Probabilistic Machine Learning - An Introduction. I'm in ...
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scipy norm.pdf return probability of a particular outcome [duplicate]
The Probability of a particular outcome is always zero, but Scipy's norm.pdf() function returns the probability value of a particular event.
For example onlinestatbook.com/2/calculators/normal_dist....
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Can someone Help me out to understand the lecturer says we're talking about Population quantiles, Population median wrt Probability density functions? [closed]
The definition of quantile given here is, Ath Quantile is at a point where the probability ( or area under distribution ) up till that point is A.
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What features can be extracted from a probability distribution? [closed]
I have been looking online regarding feature extraction and I am looking at extracting features from probability distribution by getting the characteristics of the distribution. I know that most ...
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Density function [closed]
Consider the density
\begin{align}
f(x)&=(2\pi)^{-n/2}e^{a}b\\
a &= \frac{-\sum_{i=1}^nx_i^{2}}{2}\\
b &=1 + \prod_{i=1}^n x_i e^{\frac{-x_i^{2}}{2}}
\end{align}
on $\mathbb{R^{n}}$ :
Is ...
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1answer
39 views
Gaussian Distribution: How to calculate the Cumulative Distribution Formula (CDF) from the Probability Density Function (PDF)? + Error Function? [duplicate]
I understand that we can calculate the probability density function (PDF) by computing the derivative of the cumulative distribution formula (CDF), since the CDF is the antiderivative of the PDF.
I ...
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The relationship between two probability mass function (poisson distribution)
There are two cylinder bottles with radius r1 and r2 was on the ground to collect rain drop.what is the relationship between the probability mass function of two bottle?
I guess that each of the ...
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1answer
29 views
Which assumptions should be checked for regression tree to validated model?
I am working with regression tree. I have four predictors. There is a exponential relationship between predictor and dependent variable. But after building predictive model I cannot understand whether ...
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1answer
28 views
Analyzing lie in a cardgame
We are playing a card game in which cards can be of three categories — good, bad, neutral.
A player draws a variable number of cards $n$ and then states the composition of his cards. The player does ...
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34 views
How to find outlier data points on a log-gamma distribution?
I’m dealing with a correlation network (only positive values) with M nodes where I’ve grouped features by N categories and ...
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0answers
33 views
Likelihood loss function for finite support probability distribution in Neural Networks
I have managed to reproduce solution from this article and made it work for my dataset.
Instead of making a Neural Network output a scalar (regression), we make it output two parameters of a ...
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0answers
59 views
Sum of correlated squared normals
Assume that $(X_1,X_2)' \sim \mathcal{N}((\mu_1,\mu_2)', \Sigma)$, $j =1,2$, and $Cov(X_1,X_2) = r > 0$.
We know that $X_1 + X_2 \sim \mathcal{N}(\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2 + 2r)$. ...
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Intution about the memoryless property of geometric distribution
I was watching the video about the memoryless property of geometric distribution. Here is an excerpt from the video.
$$\begin{aligned} P(X \geq x+y \mid X \geq x) &=\frac{P(X \geq x+y, X \geq x)}...
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1answer
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What distribution is $P_X(x \mid K, N) = \frac{1}{ ( 1 + N)^K} {x + K -1 \choose x} \left( \frac{N}{1+N}\right)^x$?
What kind of distribution is the following:
$$
P_X(x \mid K, N) = \frac{1}{ ( 1 + N)^K} {x + K -1 \choose x} \left( \frac{N}{1+N}\right)^x
$$
and how can I find $P_X(x < x_0 \mid K, N)$?
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What is the parameter of this Gamma posterior distribution with Poisson likelihood and constant prior?
I am trying to figure out the parameter for this driven posterior distribution.
I have searched online and found that the constant prior distribution with the Poisson likelihood function should give a ...
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The tail distribution of a binomial distribution can be expressed in terms of an appropriate beta distribution function - Explanation [duplicate]
I was reading the paper: "Estimating Probabilities of Default for Low Default Portfolios" by Katja Pluto and Dirk Tasche and they mention the following:
The tail distribution of a binomial ...
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How to interpret Hill estimate of tail index
I'm seeking a non-technical explanation of how to interpret the Hill estimate of the tail index for fat-tailed data, and, if possible, some explanation of seemingly contradictory results that ...
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Should I use Friedman's test in this case?
I'm trying to find out if certain factors matters more to people than other factors. I have five different factors and on a 5 point-likert scale, I have questions about which factor is more important ...
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2answers
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Approximating $E[g(\overline X_n)]$ and want to bound the remainder using some form of CLT or Berry-Essen Theorem
If we have a set $X_1,\dots,X_n$ of iid random variables with finite mean $\mu$ and variance $\sigma$, the CLT says that $\sqrt{n}(\overline X_n - \mu) \stackrel{d}{\to} \mathcal{N}(0,\sigma^2)$. If ...
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1answer
35 views
Seeking clarity regarding kernels
With regards to Bayesian statistics, I understand the kernel of a probability density function (pdf) or probability mass function (pmf) to be the form of the pdf or pmf in which any factors that are ...
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Some clarifications on guidelines for preparing frequency distribution of grouped data
Some of the guidelines given for preparing Frequency distribution for grouped data or "Grouped Frequency Distribution" are given below. I have some doubts regarding the same
All classes or ...
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35 views
How to find distribution of Data for building a predictive model [closed]
I am working on a project "To predict scores" . The distribution of target variable is as below :
I want to identify the type of distribution of this variable (CC$Score) .After I identify ...
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1answer
69 views
Limiting distribution of $\sum_{j=1}^{p}\lambda_j U_j$
Assume $U_j$ are $\chi^2(1)$ random variables and $\lambda_1, \ldots, \lambda_p$ are the eigenvalues of a covariance matrix $\Sigma = (r^{|i-j|})_{ij}$ with a Toeplitz-type structure (for some fixed $|...
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1answer
125 views
Does the Bhattacharyya distance and KL divergence measure the same thing?
I'm currently studying about the two methods in the title.
Does the Bhattacharyya distance and KL divergence measure the same thing, but in a different way?
I know the things that Bhattacharyya ...
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Analytical distribution of $N$ standardized samples? [duplicate]
When standardizing a set of Gaussian i.i.d. samples $\{x_i\}^N_{i=1}, X_i \sim \mathcal{N}(\mu, \sigma)$, are there analytical forms for the distributions of standardized values dependent on $N$?
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Minimum sample size required question
The lifetime (in years) of a device, $X$ can be modelled by an $Exp(1)$ distribution. What is the minimum number of devices I need so that, with probability no smaller than 0.95, at least 30% of ...
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Proof for X and Y independent if joint c.d.f. of both variables is the product of c.d.f of each of the two variables
The book I am reading says the following:
For any two variables $X$ and $Y$, if for every two sets $A$ and $B$ of real numbers $Pr(X \in A \cap Y \in B) =Pr(X \in A )Pr( Y \in B)$, then $X$ and $Y$ ...
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Biased urn experiment and Fisher's noncentral hypergeometric distribution
Let's have an urn with $m_i= 2$ balls for $c=3$ different colours with different weights $w_i$.
$n=2$ balls are taken randomly but the probability of sampling a particular coloured ball is ...
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1answer
33 views
What's the mathematical profof that kernel density estimate has the properties of a probability density?
I would like to know what is mathematical proof that kernel density has the properties of a probability density when bandwidth is greater than 0.
I know that kernel and bandwidth are nonnegative and ...
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1answer
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Metropolis Hastings with Gamma Proposal Density
I am trying to use Metropolis Hastings to sample from a shifted gamma distribution. Since it is shifted, it has a domain of $(n, \infty)$. I tried using a Gaussian proposal density and ran into the ...
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Distribution of quadratic expressions
Reading similar questions for the quadratic form of normal random variables(with the derivation here), I'm interested in how well can we generalize this approach for other distributions?
E.g., if $W = ...
