Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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Distribution of sums with multiple dice of differing sides for a probability of success. Why do distributions vary with probability?

Intro I have been looking into dice distributions within the context of rogue-like or D&D type games. I found this question. In lots of these types of games you will be attacked by something with ...
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1answer
36 views

Statistical distribution of crowdfunding donations?

Is there an accepted model for the statistical distribution of the size of crowdfunding contributions (be it simple donations or some sort of investment scheme)? Or at least some freely available data ...
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Multiplying a Normal random variable by arbitrary random variable

I'm going through "Probabilistic Machine Learning: An Introduction" by Murphy (2021), and currently trying to do Ex. 3.1 (p. 80), which states: Let $X \sim N(0, 1)$ and $Y = WX$, where $p(W ...
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Distribution of number of events that occur a certain time between each other

I'm modelling an experiment in which events happen homogenously (i.e., Poisson process). The Poisson distribution models the distribution of the number of events that occur within $t$ of a particular ...
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Marginal Distributions obtained by restricting a 2D Gaussian to a circle

Suppose I have a 2D Gaussian $$ f(x, y) = \frac{1}{2\pi\,\sqrt{\text{det}(\Sigma)}}\exp\left\{-\frac{1}{2}(\boldsymbol{x}- \boldsymbol{\mu})^\top \Sigma^{-1} (\boldsymbol{x}- \boldsymbol{\mu})\right\} ...
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Lower confidence with larger sample size? Tolerance interval for distribution-free data

I am trying to find the sample size required in order to establish a tolerance interval which contain 99% of the population, with 95% confidence. The results from Minitab for a sample size of 473 are ...
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1answer
96 views

Intuitive explanation of Friedman's H-statistic

What is the cleanest, easiest way to explain someone, a non-STEM person the concept of Friedman's H-statistic? What does it intuitively mean? While exploring feature interaction I went through ...
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RV's with same mean/variance satisying Ohlin's lemma

I am trying to find two random variables X,Y with same mean and variance such that Ohlin's Lemma holds. That is, there exists some $x_0$ such that $F_X(x) \leq F_Y(x)$ for $x < x_0$ and $F_X(x) \...
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Choosing the correct option for a symmetric distribution

I have been given the following problem and I need to choose the correct one. From the given probability description, it is clear to me that the term involving $x$ must have $|x|$ or even powers of $...
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1answer
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Why do we assign probability to theta even though we consider it constant in frequentist statistics? [duplicate]

i am trying to understand the differences between bayesian and frequentist statistics. I read that in frequentist statistics the unknown population parameter theta is considered a constant but in ...
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1answer
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Find PDF of Z=X/(Y+c), c a constant and given independence of X and Y and the PDF of X and the PDF of Y

I want to find the PDF of $Z=X/(Y+c)$ where $c$ is a constant and $X,Y$ are two independent random variables. The PDFs of $X$ and $Y$ are supposed to be given. I would like to have a general form ...
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Confused about the choice of the mean and the covariance of a function of random variables

I am very confused here. Let Z = (X,Y) be a random variable distributed according to a data distribution P_Z. Let K = XY be a random variable that depends on X and Y distributed according to a ...
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Average distance of the mean of n random complex numbers in a unit disc

Let $z_1,z_2,\dots,z_n$ be $n$ complex numbers distributed uniformly and randomly over the unit disc $x^2+y^2 \leq 1$. Let $z$ be the complex number defined by the mean of the of these numbers,that ...
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Z-scores and Probability

If $H$ is a normally distributed random variable with expected value 1.52 with standard deviation of 0.74. What is the probability that p(H=1.52)? Maybe I'm overthinking this but would you simply find ...
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Variance of $\operatorname{tr}(W^2)$ with $W \sim \text{Wishart}(n, \Sigma)$

Suppose $W \sim \text{Wishart}(n, \Sigma)$, where $\Sigma \in \mathbb R^{p\times p}$, the expectation of $\operatorname{tr}(W^2)$ is $$E[\operatorname{tr}(W^2)] =n(n+1)\operatorname{tr}(\Sigma^2) + n\...
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If $X \sim N(\mu, \sigma^2)$, then $X|a < X < b \sim \text{truncated normal}$, is it true that $a < \mu < b$?

If $X$ is normal with mean $\mu$, then $X$ in the interval $(a, b)$ is a truncated normal. However, does the mean of $X$ have to lie in the interval $(a, b)$ as well? I.e., $a < \mu < b$?
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Maximizing the difference between conditional distributions

I am running an A/B test. For each of the two tests, I get a set of samples $x_1,x_2,\dots$ each associated with features $f_1,f_2,\dots$. When I look at the distribution of values $x_1,x_2,\dots$ for ...
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Does anyone recognize this probability distribution?

Can someone tell me what distribution/ family of distributions is this?
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On the distribution of a scaled sum of a Dirchlet random variable

Consider $(X_{1},\dots,X_{K})=X\sim \text{Dir}(\alpha)$ and a vector $v=(v_{1},\dots ,v_{K})\in\mathbb{R}^{K}$. Is there a parametric density function for the distribution of: $Xv^{T}=vX^T=\sum^{K}_{i=...
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What distribution is this? [duplicate]

Basically, I am told that $\varepsilon$~$N(0,1)$, and $\omega$~$IG(\frac{v}{2}$,$\frac{v}{2})$ where $IG$ is the inverted gamma distribution Now, I am told that the distribution of: $\varepsilon(\frac{...
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Find the probability that the dial will land somewhere between $15^\circ$ and $300^{\circ}$?

Suppose you spin a dial so that it comes to rest at a random position. Find the probability that the dial will land somewhere between $15^\circ$ and $300^{\circ}$ I tried this problem but it is not ...
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Joint distribution of $Y$ and $S^2-Y^2$

Let $\{X_i\}_{i=1}^n\overset{iid}{\sim}\mathcal{N}(\mu,\sigma^2)$. Let $\{b_i\}_{i=1}^n$ be a sequence of numbers so that $\sum_{i=1}^nb_i=0$ and $\sum_{i=1}^nb^2_i=1$. Define $$S^2=\sum_{i=1}^n(X_i-\...
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Good measures to quantify the effect of outliers

I have two random quantities linked to a set of N random variables. Increasing N The two random quantities should gradually become similar (meaning that the 2 distributions narrows around the same ...
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What is this probability distribution?

Thank you in advance for any suggestions or feedback. I have a discrete 1D probability distribution represented as a vector $\textbf{p}$, $p_i = p(x_i)$. I am interested in finding the Wasserstein (...
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Estimation of the mean of the autocorrelation of the time series

I have asked a similar question before, and I was encouraged to repost the question with more details. I am not all that familiar with the jargon, so feel free to edit the post or add the tags. I ...
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Appropriate distribution for simulating a random walk between two known points, with known min/max values

I have a 1-D random walker. It starts at a value of x at time t=0. It ends with a value of y ...
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Data transformation with and without Min-max scaling

I am trying to test Tukey's Bulging Rule or Ladder of Powers (Ref Image). I discovered that I am able to reproduce the transformation (from right skewed to less right-skewed OR left skewed to less ...
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Quantitatively, how powerful is Shapiro-Wilk or other distribution-fit tests for small sample sizes?

I'm looking for an analysis (I assume a book or website, but feel free to put it all in comments here) that provides an in-depth discussion of the power/accuracy of normality assessments such as ...
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Mean and variance of likelihood function wrt data

In my previous question I asked about the distribution of the likelihood function as a random function depending on the sample of data we have. It seems that this distribution depends on the ...
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If X is Possion($\lambda$), what is the PMF of X / $\lambda$

My idea is by doing a transformation, let $X' = X / \lambda$, then $X = X' \lambda$ Since $X \sim Poisson(\lambda)$, $\displaystyle P(X'=k) = P(X = \lambda k) = \displaystyle \frac{\lambda ^ {k \...
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What are the assumptions of cross-validation if any? [duplicate]

are there any assumptions on cross-validation? For instance, is it necessary, that a) the test and training data are independent of each other? b) the distribution of the trainig data and test data is ...
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How to use Somer's Dxy and/or Spearman's p as scoring rule for interval/ordinal predictions

I am in a situation similar to this thread. I am using a Poisson distribution to calculate the probability of different ice hockey goal amounts and am using scoring rules to test their accuracy. The ...
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Closed form posterior for a mixtures of two univariate Gaussians

Giving a univariate Gaussian mixture model $$\pi_1N(x|\mu_1,\sigma_1)+(1-\pi_1)N(x|\mu_2,\sigma_2),$$ are there any priors for $\pi_1$, $\mu_1$, $\sigma_1$, $\mu_2$, $\sigma_2$ which gives a closed ...
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1answer
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Quasi Monte Carlo estimation of logit-normal density integrals

I am considering the integral $$ I(y \mid \mu, \sigma) = \int_y^1 \frac{\exp \left\{ \frac{-1}{2\sigma^2}[\textrm{logit}(x)-\mu)]^2 \right\}}{\sigma \sqrt{2\pi} (1-x)}\textrm{d}x$$ which for $y=0$ is ...
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How to properly compare two distributions

I want to compare multiple functions (one function but different parameters) which measure how much the data looks fake. Each function returns value in range [0; +inf). For my dataset with 100k ...
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Mean absolute difference for the gamma distribution

A wikipedia entry states that the mean absolute difference for the $\Gamma(k,\theta)$ distribution is $k\theta(4I_{0.5}(k+1,k)-2)$ where $I_z(x,y)$ is the regularized incomplete beta function, equal ...
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Does sample variance has a Chi-square distribution?

Let $X_1, X_2, \ldots, X_n$ be a random sample from $N(\mu, \sigma^2)$. Does $S^2=\frac{\sum^n_{i=1}(X_i-\bar X)^2}{n-1}$ has a Chi-square distribution? I know that $\frac{(n-1)S^2}{\sigma^2}=\frac{\...
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Can I approximate with a normal distribution?

I feel like I should know this (I graduated in physics a couple of years ago), but I'm really unsure about whether or not it's appropriate to use a normal distribution for the following case: I have ...
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Best books and tutorials on distribution theory

I am looking for books or tutorials to self study distributions and their properties. I know some basic about statistics and statistical inference in general, but I would like to learn more about ...
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Show that an event is improbable for exponential families iff it's improbable for all absolutely continuous distributions

Since all the exponential families are absolutely continuous, if part is trivial. However, I could not prove the only if part. My idea is to prove by contradiction, i.e. given an event $A$ such that $...
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What is the distribution of the sample variance of a chi-square distributed variable?

I am conducting an experiment, in which I have a variable that is distributed with a $\chi_k^2$ distribution (for my specific case $k=2$, but a general solution would be nice). I am taking samples of ...
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likelihood as random function wrt data

Suppose we have some dataset $x= \{ x_1, \dots, x_n\}$ where every datapoint is i.i.d., $x_i \sim P(\cdot|\theta^*)$ for some known distribution $P$ and true parameter $\theta^*$. Then, for this ...
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Minimum probability of a repitition

Let's say we have an experiment, and the outcome is described by a random Variable $X$ with sample space $\Omega = \{x_1, x_2, ..., x_n\}$. We observe the outcome of the experiment twice (the results ...
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Probability of trial sequence

A trader learns to predict whether the stock price will rise or fall on a particular day of trading. To do this, he calls one hundred friends and asks them to toss a coin once a day, thus receiving ...
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Weighted Inverse Wishart Distribution

Assume $X \in \mathbb{R}^{n \times p}$ ($p<n$). If the rows of $X$ are i.i.d. $N(0,I_p)$, we know that $$ (X^{\rm T} X)^{-1} \sim \text{inverse-Wishart}(n, I_p). $$ Let $W$ be a diagonal matrix ...
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Why is the average noise generated by the two-sided geometric distribution not null?

I am implementing the distributed differential privacy scheme proposed in this paper http://www.elaineshi.com/docs/ndss2011.pdf. Page 13 they represent a graph with the error added by a naive scheme ...
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1answer
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Asymptotic distribution of $\sum X_{i}^2$

We have $X_{1},X_{2},...,X_{n}$ as the independent standard normal random variables. Let us define: $T_{n} = \sum X_{i}^2$ then what will be the asymptotic distribution of $\sqrt{n}(\frac{T_{n}}{n} - ...
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Name these distributions

I have come across a few distributions and was wondering if they have standard names. Each of these distributions has a single parameter, $n$. Also, if you are aware of some standard reference where a ...
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Wilcoxon rank-sum or t-test with a single subject in a group

I'm just wondering if I have Group A (N=20) and group B (N=1), does it valid to do a statistical test, such as (independent) Wilcoxon rank sums or independent t-test? Note that group A and B are ...
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Basic doubts about p-values

I first summarize my understanding of p-values. We have an hypothesis $H_0$ (null hypothesis) that we want to test. Now we build a test statistics $T$, a random variable. Depending on $H_0$ being true ...