Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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2answers
128 views

Uniformly Distributed Residuals in Linear Regression

What can you say about your linear regression if the residuals are uniformly distibuted (and not normal)? I would like to consider the case I have a histogram showing residuals which are uniformly ...
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0answers
21 views

Adding Positive Skew to a Simulated Data Set

I have created code to simulate normally distributed data based on loadings from a bifactor model (Caspi et al., 2014). I need to add a skew of 2.0 to each variable in the data set to simulate the ...
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1answer
67 views

what the distribution of test statistic α as −1≤α≤1 and $f_\alpha(x) = 2\alpha x+1 - \alpha$ w/ $H_0: \alpha= 0$ and $H_a: \alpha > 0$? [closed]

For a real −1≤α≤1 , define $f_α(x)=2αx+1−α$. It is easy to see that fα is nonnegative and integrates to 1, namely is a distribution, over [0,1] . Consider the null hypothesis that α=0 , namely $...
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1answer
41 views

Relationship between distribution and data generating process

My question is: are the concepts of probability distribution, data generating process and population equivalent? If not, then what is the relationship they have. My question arises from the following ...
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1answer
24 views

Is The Jacobian Needed to Find CDF for R in Polar Coordinates?

I'm attempting to use inversion sampling to generate points on a disk according to the following PDF: $$ f(r) = \dfrac{2}{\pi(1+r^2)} $$ Here, the polar angle would just be a uniform random variable ...
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1answer
66 views

What does this Statistic mean? And how to find a density of a statistic? [duplicate]

My First Question! But it's in two parts. Context: I am given a Probability Density Function, and the question wants me to find the density of a statistic. Given pdf: $$f(x, \theta, \phi)=\frac{1}{\...
1
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1answer
25 views

Is there something like z-value for log-normal distributions?

I'm looking for a reasonable way to measure how unlikely a data point is assuming it's generated by a random variable that follows log-normal. Do we have something like Z-value for normal distribution ...
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0answers
60 views

What's a word meaning “drawn from the same distribution”?

I'm comparing results from ensemble data assimilation experiments where prior ensemble members are either drawn from the same (multivariate Gaussian) distribution as a "true" variable, or ...
1
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1answer
25 views

What family is the posterior predictive distribution in when the likelihood is a Bernoulli and the prior (and posterior) is Gaussian?

I have a problem where the hidden random variables I want to know about are continuous (i.e. my parameters can take any real number), and so I have modeled them as having a Gaussian p.d.f. (because ...
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1answer
23 views

Variance of Normal distribution given all values

I have peak value of normal distribution $0.581$ I know mean which is $0.01806$. I want to find variance now. But I know value at a certain point for continuous distribution is zero. How will I do ...
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0answers
10 views

Using asymptotic distribution to identify sequence [closed]

If I have sequences of random variables $X_n$, $Y_n$ with $X_n$ known and observable but $Y_n$ not, and I know that ${X_n}/{Y_n} \xrightarrow{d} N(0,1)$. Under what conditions is $Y_n$ identifiable ...
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1answer
36 views

Which distribution to use in the following scenario?

In a random sample of 86 athletes participating in the 2012 Summer Olympics, 63 were found to have sponsorship by private companies. Based on this sample, conduct a hypothesis test to see if there ...
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0answers
29 views

Who invented the “Histogram”?

While going through Wikipedia's article History of statistics I found In 1786 William Playfair (1759-1823) introduced the idea of graphical representation into statistics. He invented the line chart, ...
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1answer
41 views

Covariance matrix for p dimensional vector

I am working on making a conjecture about necessary and sufficient conditions for a singular covariance matrix of a p-dimensional random vector. To get to this conjecture I have to find the conditions ...
2
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1answer
25 views

Nonnormal distributions for a monte carlo simulation in Excel

My empirical (price-spread) time series data has a nonnormal distribution (see below) and it has no (partial)-autocorrelation. I am worrying about which distribution fits most to my data, but I think ...
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0answers
22 views

In comparing the exponential and Weibull distributions, why “rate” for exponential and “scale” for Weibull?

When comparing the exponential CDF with the Weibull CDF, we see: Exponential: ${\displaystyle F(x)=1-e^{-\lambda x}}$ Weibull: ${\displaystyle F(x)=1-e^{-(x/\lambda)^{k}}}$ I understand how the ...
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5answers
533 views

In comparison with a standard gaussian random variable, does a distribution with heavy tails have higher kurtosis?

Under a standard gaussian distribution (mean 0 and variance 1), the kurtosis is $3$. Compared to a heavy tail distribution, is the kurtosis normally larger or smaller?
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Estimating significance of a betting strategy, given probabilities of each event

We are given a set of independent Bernoulli trials, each having separate claimed probability (which might not be accurate). We are also given a single observation for each of those events. How to ...
4
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1answer
106 views

If all the conditionals are Gaussians, does it mean the joint is Gaussian?

Suppose I have a set of variables $(x_1, x_2, x_3)$ and I know that all the conditionals are Gaussian. That is, I know that $p(x_1)$, $p(x_2\mid x_1)$ and $p(x_3 \mid x_2, x_1)$ are Gaussian. What can ...
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0answers
14 views

How to calculate probability of values under Weibull distribution?

I have a Genomic data that shows the interaction between genomic regions that I would like to understand which interactions are significant statistically. Dataset look likes: ...
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1answer
10 views

is it the case that increasing degrees of freedom always makes every tail of a t-distribution smaller?

As per the title. Say I have X a random variable that is a 0-centered t-student. Can I affirm that P(X>a) decreases when I increase the degrees of freedom of X? Looking at the image in the ...
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0answers
21 views

How can I use transformation properties to obtain the distribution of $h(\mathbf{s})$?

Let that $\mathbf{s}=(s_1,s_2) \sim Unif(S)$, where $S$ is some spatial area. Suppose $y=h(\mathbf{s})=1-[exp(exp(\beta_0+\beta_1(\mathbf{s}-\mathbf{x})^T(\mathbf{s}-\mathbf{x})))]^{-1}$. We have that ...
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0answers
13 views

Find distribution from expected value and confidence interval [closed]

I was tasked with finding a probability distribution to fit data with an expected value of 6 and confidence that 90% of values lie between 5 and 9. I have zero experience in statistics, but my first ...
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4answers
273 views

How to generate random numbers normally distributed in R or any software with limitations (bounds)?

I am working on a project where I need to generate random numbers for a given task time which is normally distributed with mean = 40, and standard deviation = 150. Because of the high SD, I will get ...
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0answers
9 views

what is the correct way to compare log likelihood of various models?

I am trying to compare numerous neural network models in terms of log likelihood, for example there are two given here... MC Dropout - https://arxiv.org/abs/1506.02142 Deep Ensembles - http://papers....
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2answers
326 views

Poisson Gamma Mixture = Negative Binomially Distributed?

This paper introduces a model called "Beta-Geometric / NBD" which models "repeat-buying behavior in settings where customer “dropout” is unobserved: It assumes that customers buy at a ...
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1answer
20 views

Combination of distributions [reference request]

Consider a random variable $X:\Omega \rightarrow E$ that combines multiple distributions: $$X(\omega)\sim\begin{cases} N(0,\sigma^2),\hspace{0.2cm} \text{if $\omega =0$}\\ \text{beta}(0,1), \hspace{0....
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2answers
114 views

CDF of measurement vector with correlated noises

Given the following definitions: $X \sim \mathcal{N}(\bar{x}, \sigma_{0}^{2})$, and $W_i \sim \mathcal{N}(0, \sigma^{2})$, $i \in \{1,2\}$ and $E[W_1W_2]=\rho \sigma^2$. $X$ and $W_i$ are independent. ...
5
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2answers
142 views

Distribution of the inner product between a noise-free and a noisy signal

I am working on a problem where we have a noisy measured signal, which is stored as an $N$-dimensional vector $\mathbf{Y},$ and a set of $n_s$ simulated noise-free signals $\{\mathbf{X}_i\}_{i=1}^{n_s}...
3
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2answers
95 views

finding out distributions mathematically

we know that the larger the degree of freedom, the less likely extreme events will occur (e.g., if you throw fair coin once, odds of heads is 50%, if you throw twice, odds of two heads are 25% and so ...
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24 views

Normal Distribution problem [closed]

An underwater salvage team is preparing to explore a site in deep sea where 45 ships sank. The company feels these wrecks will generate an average rs 2250000 and sd rs 390000. The finance head of the ...
14
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2answers
432 views

What is the origin of the name “conjugate prior”?

I know what a conjugate prior is. But I'm confused by the name itself. Why is it called "conjugate"? A complex conjugate $z^\ast$ has a reciprocal relationship with $z$, i.e., ${z^\ast}^\ast ...
2
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1answer
68 views

Sampling uncertainty of posterior probability distribution

I'm working on a problem with 3 possible outcomes and a bunch of features. I have a regression model that outputs probabilities for each category and I'd like to extend these probabilities to ...
4
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2answers
38 views

Why would you want to fit/use a Poisson regression instead of Negative Binomial?

Given that Poisson is a special case of Negative Binomial which seems to just make error more likely in the case of overdispersion, without offering any real benefits, why would you fit a Poisson ...
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0answers
9 views

inequality involving mean, median and variance [duplicate]

I'm looking to show $|{\rm med}(x)-\bar{x}|\le{\rm sd}(x)$. I did a bunch of simulations and the statement seems right to me. $$ {\rm Var}(x)=\frac{1}{n}\sum\left(x_i-\bar{x}\right)^2=\frac{1}{n}\sum\...
3
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1answer
48 views

(Non-limit) distribution of maxima from different univariate, discrete and stationary time series

Motivation: I'm currently studying the convergence of maxima from simulated time series to max-stable distributions, and in order to do so, I want to better understand the penultimate distribution of ...
2
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1answer
27 views

Should I use highly skewed features in my model?

Many of my features are highly skewed as you can see below in the figure. Should I be using such skewed data for modeling purposes? If I cannot, then is there any way to integrate such features in my ...
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0answers
23 views

guessing a number between 1 and 100

Person A chooses an integer between 1 and 100 at random, then B can guess that number in (at most) 7 attempts, i.e. $\log_2(100)+1=7$. What if now A chooses an integer from a distribution that is ...
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0answers
12 views

Uniformly Most Powerful (UMP) statistic for N Poisson distributions

For samples of $M$ Poisson-distributed datapoints $X_{1, r},...,X_{M,r}$ $\sim$ $Pois(\lambda_{r})$, and $N$ such distributions ($1 < r < N$), I have defined a likelihood function to describe my ...
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0answers
15 views

What model is a suitable model for zero-constrained variables?

After reading the answer in How do I normalize a bimodal distribution?, I began to think about the concept more broadly. With OLS on zero-constrained variables, the residuals signal bias, as the zero-...
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1answer
25 views

When you do a random permutation F test (by permuting group membership) is inference made on the samples or the populations?

I am trying to understand different approaches that use randomization procedures. One thing I cannot come to a clear conclusion on is this: Say we have randomly sampled individuals from 2 populations. ...
3
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2answers
139 views

$X_{1},X_{2},X_{3}\overset{i.i.d.}{\sim}N(0,1)$, find m.g.f. of $Y=X_{1}X_{2}+X_{1}X_{3}+X_{2}X_{3}$

I tried this $X_{1}X_{2}+X_{1}X_{3}+X_{2}X_{3}=X_{1}(X_{2}+X_{3})+\frac{1}{4}(X_{2}+X_{3})^{2}-\frac{1}{4}(X_{2}-X_{3})^{2}$ $U=X_{2}+X_{3}\sim N(0,2)$ $\psi_{X_{1}(X_{2}+X_{3})}(t)=\psi_{X_{1}U}(t)=\...
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0answers
30 views

Covariance matrix of integral of multivariate normal distribution

If $t = [t_0, t_1, \dots, t_{N-1}] \in \mathbb{R}^N$ with $t_i \sim N(\mu_i, \sigma_i^2)$ and its covariance matrix $C \in \mathbb{R}^{N \times N}$ where $C_{ij} = Cov(t_i, t_j)$ is given If I define ...
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0answers
27 views

Quantile Matching using the skewed t-distribution from Azzalini & Capitanio (2003)

I try to replicate some findings from a paper (page 11/12 of https://www.newyorkfed.org/medialibrary/media/research/staff_reports/sr914.pdf). In this paper they estimated some quantiles using quantile ...
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0answers
28 views

Finding variance and mean of an expression?

I want to compute the expectation and variance of the following expression. $ Z|H_{i}=(|h_{i}|^{2}|s[n]|^{2}+|w[n]|^{2}+2\mathcal{R}(h_{i}s[n]w^{*}[n]))$ where,$s[n]$ is PSK signal and $h$ is rayleigh ...
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0answers
33 views

How to define and plot a distribution function in python?

I want to define a distribution function (gaussian or skewed,...), the X axis is from 0 to 255. I have the mode which is located at the point 100 and i have two points (40, 170) that i consider ...
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0answers
21 views

Are segments painted randomly respective to previously painted segments?

I have a sequence of $N$ binary values (+ and -). At $t=0$, all values are minuses (-) ...
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0answers
22 views

Are there convenient methods/tricks to make calculations with non-independent terms? (two examples here in particular)

In a recent question it came out that I needed to calculate the sample distribution of $\dfrac{Cov(XY,X)}{Var(X)}$ where the distributions of $X$ and $Y$ are known. By this, I mean the law of $$\dfrac{...
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0answers
12 views

Testing a distribution vs. random and subsampled distributions

I am willing to test the distribution of a sample against another one. The problem is that the other distribution is randomly taken as a subsample of a bigger sample. I tried with a simple wilcoxon ...
-1
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1answer
26 views

Finding C for a PMF of a frequency distribution

N has probability mass function: $p_o = p_1 =0$ and $p_k = c/k!$ for $k=2,3,4,...$ I used exp series $\sum_{n=1}^{\infty} \frac{x^k}{k!} = e^x$ to get $ c\sum_{n=1}^{\infty} \frac{1}{k!}$ then $ce=1$ ...