Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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46 views

Joint asymptotic distribution and measures of fit

I have been looking for something similar but couldn't find any, unfortunately. Some help would be much appreciated. Consider a simple bivariate linear mean regression $$y=\beta x + e$$ where $E[e|x]=...
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66 views

Does there exist a universal random variable which can represent all events in a probability space?

Given a probability space $(\Omega, F, P)$, I know that for each event $A \in F$, we can define an indicator random variable $I_A$, then $A$ corresponds to $I_A=1$. I am wondering does there exist a &...
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1answer
49 views

Statistical tests for distributions on constrained intervals

I have a continuous distribution of angular values on the interval of 0 to 90 degrees. This distribution is expected to follow a half-normal distribution with a greater frequency of values toward zero....
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Binomial Hypothesis testing finding p value

Question: how to find a p-value where: $H_0: \hat{p} = 1/6 $ $H_a: \hat{p} \ne 1/6 $ Significance level is $\alpha=.05$. In my test: $n = 60$, $x = 14$, so my $\hat{p}=0.2333$. How do I find a p-...
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How to understand the difference between the mixture of same distributions vs. convolution (sum) of random variables of same distributions?

Before I ask the question, let me introduce how I came to this problem. Recently I learned about the linear regression. It was said, that the residuals of the model should be normally distributed. We ...
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Measuring distance between distributions with different kernels functions

currently I am using maximum mean discrepancy to measure distance between multivariate distributions. The related reference is like An explicit description of the reproducing kernel Hilbert spaces of ...
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32 views

Consistency and asymptotic distribution

This problem is from Hansen's econometrics textbook, chapter 7. The model is $Y = X'\beta+e$ with $E[e|X] = 0.$ An econometrician is worried about the impact of some unusually large values of the ...
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Find control chart limits based on existing data

I'd like to produce a control chart that tells me if a given process will be within bounds in the future or not. Currently, the process gives me simple timeseries data, per minute ...
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Sampling from distributions with a rules filter

I want to generate random numbers from different distributions such that they always satisfy certain rules. The idea is that two variables can be randomly sampled from respective distributions, but in ...
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7 views

Which assumptions and constraints are relevant in a mixture model to use either a Gaussian, Weibull, Gumbel or other distribution?

For a dataset that I'm analyzing, I can obtain a series of distributions on a given feature space, and I can assume that they take the form of collections of uni-modal clusters (think of something ...
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23 views

Averaging the parameters of multiple weibull models

I have multiple different weibull models whose parameters have been estimated on different populations. I would like to "average" those different models into one model based on my beliefs of ...
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1answer
29 views

Bayesian Estimation of CDF

i'm getting pretty confused by the following problem, hope anyone can clarify my mind: Using a bayesian approach obtain a posteriori and interval estimations for $\mathbf{F}_{X}(x)$ using a Uniform(0,...
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1answer
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How to intuit the covariate shift?

Out of distribution and shifting data distribution are two types of dataset shift 1, I can understand what out-of-distribution means but not what shifting data distributions are. In that blog an ...
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135 views

Source recommendation for statistical modeling: theoretical and practical aspects

I have asked following question few days ago. Fitting distribution: Covid-19 confirmed cases It turned out that I have a deeper problem beyond the statistical issues that I am dealing with. I didn't ...
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19 views

Repeated Poisson draws - stationarity

Many individuals (say, $N=500$ or more) independantly draw from a Poisson distribution with mean $\lambda$. This gives the "Time until next update", $t_u$, for each respective individual. At ...
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understanding p value and distribution [duplicate]

I am from a biology background. Using t, χ2, F tests day-to-day, following like a recipe. However, I feel I must understand the background of this. I took an online lecture on p-value and hypothesis ...
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Distribution of $(X_1-\overline{X})/S$ where $X_i$ are i.i.d. normal [duplicate]

I have encountered the statement says that the statistics $${{X_1-\overline{X}} \over {S}},$$ where $X_i$ are i.i.d normal, is ancillary (where $S$ is the sample variance square root). I know that $(...
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1answer
24 views

Finding dynamic probabilities in R

I have a sequence of independent uniform random variables $X_1,X_2,X_3,... \sim \text{IID U}(1,3)$ and I need to compute the value: $$\sum_{n=1}^\infty \mathbb{P} \Bigg( \sum_{k=1}^n X_k < a \Bigg)....
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Maximizing likelihood function of continuous normal distribution [duplicate]

If I wanted to maximum likelihood estimator for $f(x)=\frac{1}{\sqrt{\sigma^22\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$ where $-\infty\leq x \leq \infty$, my original plan was to do $\Pi^{n=\...
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Quantifying if two datasets are from the same distribution, if I only have distances

Let's say that I have two datasets. The target dataset is 1K instances, the "predicted" dataset is 1M instances (but I could downsample). I can compute the distance between instances. How do ...
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1answer
30 views

Why Do Distributional Forecasts Need to Produce Normally-Distributed Forecasts to be Ensembled/Combined?

I am forecasting a collection of different types of items, using many different forecasting techniques. Some of the techniques I use take the input data as is to produce a distributional forecast. ...
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Is there a closed form solution for the Hellinger Distance between two Wishart distributions?

Given two Wishart distributions $X_0 \sim W_{p_0}(V_0,n_0)$, $X_1 \sim W_{p_1}(V_1,n_1)$, what is the Hellinger Distance between them? Can it be obtained in closed form assuming $p_0 = p_1$? For ...
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Calculating a covariance [duplicate]

Consider two stochastic processes: $T_t$ is the number of times we flip a coin with bias $\rho$ by time $t$, and $X_t$ is the number of heads of the coin by time $t$ (i.e. after $T_t$ flips): $$T_t \...
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1answer
27 views

probability of left handedness [closed]

13 random people were chosen from the US population . 12 were found to be left handed, (none ambidextrous). Assume the incidence of left handedness in the US is 13%, and does not change with gender ...
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Dividing two gamma distributions [closed]

I am trying to determine the distribution function of 2 random variables X and Y such that X/Y. I have determined that X~gamma with parameters alpha = 3/2 and beta = 4 and Y~gamma with parameters ...
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1answer
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Multiplying a chi-square distribution by a constant

If $X\sim\chi^{2}(3)$. What is the distribution of $2X$?
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1answer
31 views

Should I chose a linear, a generalised or a mixed model?

I have a dataset of n=3000 nested within 8 countries with approximately 200 or 400 responses in each country and planned to perform multilevel modelling with 4 dependent variables (DV) as fixed ...
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Sampling distribution of loss function

So I believe the sampling distribution of the likelihood function is a basic idea in frequentist statistics. For example, the Fisher information $\text{Var}_x(\nabla_\theta \log P(x|\theta))$ which ...
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29 views

Distance to uniform distribution for continuous probability distributions [closed]

I need to quantify the distance of a continuous probability distribution $f(x)$ to the uniform distribution on $x \in (-\infty, +\infty)$. Do you know any distance functions that could help here? For ...
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2answers
109 views

Maximum Value of Kernel Function in ABC

Are there cases where a kernel function, must have 1 as the maximum value ?? The definition of a Kernel can be found in the following link, https://en.wikipedia.org/wiki/Kernel_(statistics)#In_non-...
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Binomial-like distribution which the probability changes after certain trials

Let $0<p<q<1$. Consider a trial with success probability defined below. If it is a $100, 200, 300$th trial after the last success, the probability is $q$. If it is a $400$th trial after the ...
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Validity of sequential testing of parameter-eliminating hypotheses on a multi-parameter distribution

Suppose I have a continuous univariate distribution f(x: θ) where θ is a length-n vector θ = (θ1, θ2, ... θn). Suppose I fit such a distribution to a data set, and I want to test whether a simpler ...
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Replace missing predictions with actual values altered by an error

I have two time series: One year of the actual values of Y My predictions for Y in that year made by someone's guess The second time series, the one with the predictions, is missing two months. I ...
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Are there really many continous distributions without the PDF [duplicate]

This is one interesting question I take some time to search if there is any distribution function that is continuous but without the PDF. After some search I found Cantor distribution, sometimes ...
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1answer
50 views

I am measuring airborne bacteria in a new high-risk room; how do I determine sample size?

I am measuring airborne bacteria in a new high-risk room. The bacteria are measured with Petri plates that have area 19,5 cm$^2$. I have 30 old measurements results, that give average bacteria counts ...
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What theoretical distribution could this data be coming from?

I am trying to build a regression model to predict a variable whose count histogram and QQ plot is given blow. Does anybody have any idea know what theoretical distribution could this variable be ...
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1answer
121 views

PDF given distribution function

I am given a distribution $$F(x)=\frac{1}{2}+\frac{x}{2(1+|x|)}$$ and I need to find the pdf. I did the following- $$f(x)=F'(x)=\frac{2(1+|x|).1 -2.sgn(x).x}{2(1+|x|)^2}=\frac{2+2|x|-2|x|}{2(1+|x|)^2}=...
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Why is the normal distribution used in linear models, but in generalized linear models the exponential distribution is used?

Why is the normal distribution used in linear models, but in generalized linear models (GLMs) the exponential distribution is used?
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Distribution of players in a computer game on different “levels”

Many computer games have different levels. Here is a description about levels on wikipedia, but I will explain more precisely what I mean below. The idea is this: to optimize the experience with a ...
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0answers
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Computing a probability involving several probability distributions [closed]

I am currently performing the following procedure: Generate $n$ i.i.d. random variables $a_1, a_2, \ldots, a_n$ according to a $U(0, L)$ distribution for some fixed positive constant $L$. Consider a ...
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Standardizing Data to make it relative to all periods rather than portions

Inputs: I have 5 years worth of google trends data daily relative to a 3 month periods (Jan-March,April-June,etc.). Ex: Jan 1, 2020 and Feb 8, 2020 are both relative to Jan 1,2020-March 31,2020 from ...
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Mean number of throws to exceed a threshold [duplicate]

Say that you have a die with n faces, and you need to throw the die until the sum of your results exceeds a given threshold. What is the average number of throws needed? I think that to compute that ...
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1answer
49 views

Can these two situations be efficiently distinguished? [closed]

We have N+1 dice, one two-sided and N 6-sided ones. The marginal probability density functions are such that each die is fair, i.e. each face shows up with probability $1/6$. We want to distinguish ...
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2answers
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Finding Probabilities of Normally Distributed Random Variables [closed]

A random variable $x$ is known to follow a normal distribution with mean $35$ and standard deviation $7$ Find the following probabilities: a. $P(x<25)$ b. $P(x<33)$ c. $P(x>42)$ d. $P(x>35)...
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2answers
107 views

Integral of cdf times pdf is a probability?

Let $X$ be a random variable with distribution function $F_X$. Consider $$P=\int_0^\infty (1-F_X(x))e^{-x}dx.$$ Because $1-F_X(x)$ is the probability of $X>x$ and $e^{-x}$ is the pdf of an ...
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1answer
73 views

A distribution like lognormal, but limited from two sides

I would need a probability distribution which reflects the following user input: Value 0.5...9.5 Belief into that value The higher the belief, the more the distribution is Dirac-like. The lower the ...
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1answer
30 views

finding probability distribution of sum of 2 random variables

I have a probabiliy distribution $$p(x) = \begin{cases}e^{-x} & x\geq0\\ 0 & x<0\end{cases}$$ I need to find the probability distribution for $Z=X+Y$ where X and Y are from the above ...
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1answer
65 views

Probability of sample mean

Consider independent random variables $Z_{i}\sim N(0,1),\;i=1,\ldots,16$ and let $\bar{Z}$ be the sample mean. Then calculate $P\left[Z_1-Z_2<2\right]$ This is an example of one of the ...
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0answers
23 views

Find the following summation quantity for a Poisson Distribution [closed]

If I have a random variable $X$ with Poisson distribution with parameter $n\theta$, i.e. $P(n\theta)$, then it is required to compute the following quantity. $$\frac{e^{-n\theta}}{(n-2)^3}\sum_{x=0}^{\...
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1answer
43 views

Sum of squared normals: $ \sum_{k=1}^{n}U_k^2 = \frac{1}{n}\sum_{k=1}^{n}a_k^2 \cdot \Gamma\left(\frac{n}{2},\frac{1}{2}\right)$

Assume $U_k \sim \mathcal{N}(0,a_k^2)$, where $a_k \rightarrow c > 0$ as $k \rightarrow \infty$. It follows that $U_k^2 \sim \Gamma(\frac{1}{2}, \frac{1}{2a_k^2})$. I'm interested in the exact and ...