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Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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PDF transformation for many to one function [duplicate]

I would like to find the PDF of the random variable $Y$ given the PDF of $x$. $$Y=sin(x)$$ $$f(x) = 2x/(pi^2) for 0<x<pi$$ and 0 otherwise. Following the tips in the question here: https://...
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When is the pmf of the difference of two independent random variables symmetric in zero?

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 (\...
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1answer
142 views

Distribution of sum of independent exponentials with random number of summands

Let $\tau_i\sim\exp\left(\lambda\right)$ be independent and identically distributed exponentials with parameter $\lambda$. Then, for given $n$, the sum of these values $$T_n := \sum_{i=0}^n \tau_i$$ ...
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2answers
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What can we say about $N_{i}$ where $N=N_{1}+\cdots+N_{m}$, $N\thicksim Geom(\frac{1-p}{p})$ and conditional distribution of $N_{j}$ is binomial

Suppose that the number of events $N$ is a Geometric random variable with mean $\frac{1-p}{p}$. Further suppose that each event can be classified into one of $m$ types with probabilities $p_{1},p_{2},\...
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1answer
30 views

How to generate a conditional random variable in R? [on hold]

Suppose there is a sample $X\sim N(0,1)$ x<-rnorm(100). If I want to generate a conditional random variable $Y|X\sim U(0,1)$, how can I get this conditional ...
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Is there a general analytical form for marginalizing this set of distributions?

Let $f :\mathbb{R}^{2D} \to \mathbb{R}^{2D}$ be a bijective function. Let $x \sim N(0,I), v \sim N(0,I)$ both are dimension $D$ random variables. Now joint random variable $(x,v)$ is a Gaussian ...
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1answer
23 views

log skew normal computation

I want to compute probabilities assuming data have log normal and log skew normal distribution (in R). As I couldn't find any package that directly computes log skew normal (as plnorm does log normal),...
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Calculating error/confidence interval on points of a distribution

This is a beginners question, coming from an absolute beginner in statistics, but I haven't been able to find anything about this online, probably because I don't even know where to start looking. My ...
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6 views

Comparing groups of empirical probability density functions

I'm a neuroscientist approaching doing some complex statistics and can't find anyone to point me in the right direction of what I need. I'm happy to read around the subject, I just don't know enough ...
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Linear Regression: Calculating a treatment effect directly in regression vs. averaging potential outcomes

Suppose I have the following true model, where an individual $i$ at a particular point in time $t$ is either treated ($W=1)$ or untreated ($W=0$). The outcome for individual $i$ at time $t$ under ...
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How to create a model that will display a graph/table showing predicted units sold?

Goal: Create a model that will display a graph/table showing predicted units sold and then be able to use that to compare actual units sold during the season, eventually turning into an accumulated ...
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28 views

How to compare two dispersion measures in a specific probability density function?

I am very interested in the properties of a specific probability density function (pdf) proposed in this article. It seems to me that this pdf is a very general one to capture the properties of wide ...
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1answer
21 views

How to modify the mean and variance/dispersion of a given distribution

I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution. Ideally, this adjustment would be implemented through a parametric function ...
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13 views

What method to use for conditional density querying

I have a dataset of 3d poses each represented by 40 points (all relative to the central point). So my data has dimensionality 120. What is needed is to learn how build realistic pose, when positions ...
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1answer
45 views

A random variable with a step function CDF is discrete?

Consider the step function $$ \Delta(x;\lambda,\mu)\equiv \sum_{j=1}^J \lambda_j\times 1\{\mu_j\leq x\} $$ where $\lambda_j\geq 0$ $\forall j$; $\sum_{j=1}^J \lambda_j=1$ $\mu_j\in \mathbb{R}$ $\...
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metrics or distributions in high-dimensional spaces?

This is a somewhat naive question that occurred to me while thinking about organisms that potentially have to deal with high-dimensional sensory input. I wonder which problem is harder in theory: ...
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Conditioning the probability obtained from a machine learning model

I have developed a random forest classifier to predict whether a customer will churn. The data used to produce this model has the following form ...
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Is the uniform distribution stationary? [on hold]

Is the uniform distribution stationary or non-stationary? The normal distribution is stationary in all cases? The normal distribution is non-stationary in which cases? or in all cases else normal? -...
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Sampling Distribution of sampling proportion [closed]

A population consists of N=6 numbers 1,3,6,8,9 and 12. Draw all possible samples of size n=3 without replacement from the population and find the proportion of odd numbers in the samples.
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Distribution of a square of a random variable [duplicate]

If X is normally distributed with mean mu and variance sigma^2, then what can I know about the distribution of X^2 ?
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3answers
57 views

ANOVA and binomial distributions

I am really struggling with this question: Given the assumptions for t-tests and ANOVAs, should data generated by a binomial process be analyzed with them? Obviously, no. However we ran several ...
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29 views

proof related to gamma distribution function [closed]

Finding gamma distribution from the given probability distribution function as shown in this picture. A random variable x is said to be gamma-distributed with index α if its probability density ...
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1answer
48 views

Variance of unbiased estimator for the shape parameter of Pareto distribution

I'm interested in getting the error bounds of the unbiased estimator of the shape parameter ($\alpha$) using maximum likelihood method of Pareto distribution. The unbiased estimator is known to be ...
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Plotting groups of sample distributions as CDF with dispersion

Context I repeatedly run an experiment each repetition of which results in a sample of observations that I assume follow some distribution. (It is not important here what the distribution represents.)...
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Can I ignore statistical distribution if errors are small

I'm struggling to find the right distribution for my data and only after I know which distribution suits best I wanted to select a certain statistical model. But now I can't find an appropriate ...
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1answer
42 views

Distribution of the $L^{2}$ norm of a vector of components drawn from Gaussian distributions

I recently asked this question involving uniform distributions. I am wondering what would be the equivalent for Gaussian distributions. The problem states as follows. We consider a random vector $\...
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1answer
58 views

How to statistically analyze the relationship of right skewed data

I struggle to analyze these continuous data: The last four plots show the diagnostic plots on my model (model <- lm(data 1 ~ data 2). My aim is to investigate the relationship between data 1 and ...
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1answer
40 views

Are “the probability of something being seen” and “the probability of someone seeing something” the same?

I'm puzzling over how to treat the distribution of TV advertising spots. I have a list of spots and the % of the population who saw them. What I'm trying to get is a Gamma distribution representing ...
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0answers
47 views

Correlated Bernoulli Trials

Suppose there are $n$ dependent Bernoulli trials, $X_{1}$,...,$X_{n}$ with $% X_{j}\in \{1,0\}$ and $\Pr (X_{j}=1)=q$ for all $j=1,...,n$. For any $% n\geqslant 2$ dependent Bernoulli trials, in the ...
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2answers
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If $X \sim a\chi_1^2$, and $Y \sim b\chi_1^2$, what's the distribution of $Z = X/Y$?

Assume that $X$ and $Y$ are independent. $Z = \frac{X}{Y}$ a ratio of independent $\chi^2$ random variables with 1 degree of freedom. If $X \sim a\chi_1^2$, and $Y \sim b\chi_1^2$, what's the ...
3
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1answer
103 views

What would be the output distribution of ReLu activation?

Suppose my data has a normal distribution and I am using an NN as a model, wherein I am applying ReLu, non-linearity to it. I am curious to know how the output distribution of the ReLu looks like? ...
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1answer
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Find maximum likelihood given Rayleigh probability function

Problem Suppose we use a Gaussian PDF to express the likelihood of light intensity prevalent on Clear, Cloudy, and Eclipse weather. The probability of a certain amount of light value (positive or ...
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What scenario corresponds to choosing the “true distribution” $p$ in $\textsf{KL}(p\parallel q)$?

I understand that when you think about changing $q$ in the Kullback-Leibler divergence $\textsf{KL}(p\parallel q)$, this corresponds to trying to find the distribution that minimizes information loss ...
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24 views

Beta-binomial distribution for scaled and translated Beta

Recall, that a binomial distribution in which the probability of success at each trial is randomly drawn from a beta distribution results in the so called beta-binomial distribution. One can calculate ...
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31 views

Which properties yield the exponential family of distributions?

It seems like every resource that discusses exponential families simply defines the family of distributions, explains why it's useful and then derives some of its properties. I have only seen one ...
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0answers
45 views

Must the domain of a CDF be $\mathbb{R}$ or can it also be a strict subset?

So my question is whether the domain of a cumulative distribution function has to be $\mathbb{R}$ or whether it can also be a strict subset. The reason I'm asking is because I'm currently going ...
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Why isn't there a probability distribution with the CDF of the form $bx^a$? [closed]

I'm trying to fit my univariate continuous data $x \in (0, \infty)$ to a simple parameteric model for the purpose of simulating additional samples. The log-log probability plot is linear and (mis-)...
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0answers
20 views

Confidence Interval of shifted exponential

. I know what confidence intervals are but can a confidence interval of given size have different lengths?
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0answers
27 views

How to plot Gaussian Likelihood with unknown mean?

Plot gaussian likelihoods with unknown means, conjugate priors of the means and their corresponding posterior distributions. I'm a beginner to statistics and its distributions. Trying to figure out ...
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1answer
27 views

Why can we treat MGF in this way

For the standard proof that if $Z \sim N(0,1)$ than $Z^{2} \sim \chi^{2}_{1}$ We write: $$M_{Z^{2}}(t)=\int_{\mathbb{R}}\exp(tz^{2})\frac{1}{\sqrt{2\pi}}\exp(\frac{-z^{2}}{2}) dz$$ That is, we use ...
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0answers
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Probability mass function of n dices thrown twice [duplicate]

You roll n dices independently from the other rolls. Than you roll again the dices that show a number other than 4. X = number of 4s. a. What is the kind of the distribution of X? What are the ...
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0answers
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How to see this order statistic result and find my error

Let W be a random variable with pdf $f(w)=\theta B^{-\theta}w^{\theta-1}$ for $0 \lt w \lt B$ and 0 otherwise. Assuming Independence, Show that , $W_{n:n} \to B$ as $n \to \infty$ where $W_{n:n}$...
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0answers
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Other than linear transformations of any one Bernoulli distribution, what popular distributions only take two possible values?

I suspect that this may be a silly question, as some sort of isomorphism ought to exist between linear transformations of the Bernoulli distribution and any other distribution that can only take two ...
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0answers
69 views

Maximum likelihood estimators of $\theta$ in $U(2\theta-1,2\theta+1)$ distribution

I understand why (D) is one of the answers but i dont know about the rest?
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1answer
23 views

How can I discriminate between mesurement error and real change working with data expressed as percentages?

I am analizing cuantitative changes in estrogen receptor (ER) expression in paired breast cancer biopsies (biopsies of the same patients separated by time T). ER expression is measured as a percentage ...
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Distribution of the $L^2$ norm of a vector of components drawn from uniform distributions

We consider a random vector $\vec{v} = \left(x_{1}, x_{2}, \dots, x_{n}\right)$ built from $n$ real random variables drawn from a real continuous uniform distribution $\mathcal{U\left(a, b\right)}$, $...
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1answer
50 views

Which statistical methods are best suited for distribution with two peaks?

My data shows this distribution: I am looking for a statistical distribution which my data follows. Thought about poisson distribution, but goodness of fit test shows p < 0.05
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1answer
36 views

The probability density function of half-chi-square distribution

Let $X$ be a random variable from a chi-square distribution with 1 degree of freedom. The probability density function (pdf) of $X$ is $f(x) = \frac{\exp{(-x/2)}}{\sqrt{2\pi x}}$, $x>0$. In the ...
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0answers
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benchmarking a right skewed distribution

I'm benchmarking procedure groupings done in a hospital and all the items associated with the cost of the procedure. As you can imagine the range can be from .01 to 50k dollars per item. With the ...