Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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The multivariate Inverse-Gamma

On wikipedia they give a multivariate form, which to my understanding is used when V is known up until the scaling factor for a Normal-InverseGamma conjugacy. I tried to find a source of the ...
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Negative multinomial distribution

The pmf of negative multinomial distribution is \begin{align*} P(\boldsymbol{\rm{X}}=\boldsymbol{\rm{x}})=\frac{\Gamma\left(x_0+\sum_{i=1}^{m}x_{i}\right)}{\Gamma\left(x_0\right)}p_0^{x_0} \prod_{i=1}...
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Hazard function and survival analysis

I have a function $\lambda(t)$ which returns the instant probability of dying at a time $t$. I'd like to compute the function $F(t)$ which returns the probability of being dead at a certain time $t$. ...
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Monte Carlo simulation or Bootstrap for determining sample L-moments estimates

I'm trying to estimate the sample L-moments of stations from the Annual Maximum Series of precipitation. The book by Hosking (1986a, 1990) recommend using Monte Carlo simulation in generating ...
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How to fit a GLMM with non-negative non-integer continuous data?

I am trying to analyse a data set which has numbers of behaviours performed under 2 different predictor variables. The numbers of behaviours were standardised to rates per minute, since the ...
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Calculate the likelihood at the mode of a pdf conditioned on θ [closed]

Assuming that I have the PDF of a random variable X with parameter θ (i.e., f(x|θ)). How is the likelihood at the mode of this PDF computed?
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What is the distribution of the daily change rates of S&P 500?

I was wondering what is the distribution of the daily change rates of the S&P 500. It doesn't seem to be distributed normally (using qq plot for example).
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Limit distribution of the joint distribution of maximum and minimum of a sequence of random variables

Assume we have a sequence $\mathsf{X}_1,\mathsf{X}_2,\mathsf{X}_3,...$ of iid random variables. Then the Fisher-Tippet-Gnedenko theorem shows that $$ \mathbb{P}\left(\frac{\max\{\mathsf{X}_1,\mathsf{X}...
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classifying into 3 groups and measure distribution differences

I'm meeting an issue that I've tried to describe as clearly as possible though I'm sure it would benefit to be reframed in statistics wording, especially the title. Would be grateful to get some ...
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Find pdf of X+Y [duplicate]

Let X ∼ Exp(λ) and Y ∼ Exp(μ) be two independent exponential random variables, where λ, μ > 0. Find the probability density function of X + Y if λ ̸= μ. I have successfully find ans if λ = μ, but ...
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Binary Classification Problem with Predicted Probabilities distribution skewed

I have a balancedrandomforest model which was trained on unbalanced data (92/8) for a binary classification problem. The AUC is around 0.98 and the precision and recall are also acceptable being 0.89 ...
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Betting on a sample from a known distribution

This was an interview question. Given a known distribution, sample a value from it with replacement for many times. Two people A and B bet on the sample with their own guesses, and the one closer to ...
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"Consensus" on Analyzing Mixed Continuous and Categorical Data in the field of Statistics? [closed]

I have been trying to determine the popular "consensus" as to how mixed continuous and categorical data (e.g. a dataset that has variables on income and gender) is generally analyzed in the ...
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how to generate data from beta-Liouville distribution?

I want to test my model following the beta liouville distribution, so as a synthetic data, I need generate data from this distribution. can anyone mathematically tell me how to calculate it? this is ...
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Find asymptotic variance of the moment estimator

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $$f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$$ and we ...
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If Y is the sum of normalized multivariate pareto random variables then Y is a Feller-Pareto random variable

If we let $\underset{k\times 1}{\boldsymbol{X}}=(X_1, \dots, X_k)' \sim MP^{(k)}(\boldsymbol{0},\boldsymbol{1}, \alpha)$ where MP denotes a Multivariate-Pareto distribution, with joint survival ...
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Skewed distributions: choosing the median or average

Goal: To determine whether to use the median or average (or a weighted combination of the two) from a data set based on whether ...
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Usefulness of KS tests and other similar distribution comparing tests

I am working on a machine learning binary classification problem. I have an outcome variable status called as loan paid and <...
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QQ Plot help meaning [duplicate]

How can I interpret the following QQ Plot? Can you explain it for example for the point 20 and 12?
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If X follows the distribution with pdf $f\left(x\right)=\theta x^{\theta-1}\ $, prove that the distribution of $\mathrm{\Sigma}\log{x_i} $ is Gamma [closed]

I am trying to get log(x) to be exponential but I can't do it. Any help will be appreciated.
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2 votes
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Prove $P(X_1+X_2> 2C) \leq P(X_1>C)$ if $X_1,X_2$ are identical, but dependent?

If $X_1,X_2$ are dependent but identically distributed, it seems obvious that $P(X_1+X_2\geq2C) \leq P(X_1\geq C)=P(X_2\geq C)$. At least if we additionally assume that the joint distribution is ...
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Can you help me with distribution/Skewness

Can you tell me if the data normal distributed are? It looks a bit strange. I would say it is skewed 40,80 and so on are sizes of the Apartments in m^2
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Distribution of the Spearman rank correlation coefficient under the assumption of non-zero correlation

There are some papers and some R packages providing exact calculations for the CDF and the inverse-CDF of the (sample) Spearman rank correlation coefficient. My question is: How difficult would it be ...
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I tried to solve this problem but got stuck in the end - Sampling distribution

This is a problem about sampling distribution that got me a little confused.. The time it takes for students to complete their university degrees has a distribution normal mean 6,4 years and standard ...
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Methods for modelling distributions?

As predictor X I have particle size distributions and I would like to run a model y ~ X. I.e. each trial has a response ...
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What distribution to sample X from to get an uniform distribution in Y?

I have a random variable $X$ which is related to another random variable $Y$ as $Y = \text{cos}(X)$, where $X \in [0, \pi/2]$, and I would like to know what distribution I should sample $X$ from in ...
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How to convert percentage values from 7 point scale to 5 point scale?

Suppose in one year I had a survey with 7-point scale and the values are like ...
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Deriving the CDF of student $t$-distribution

I am trying to derive the cumulative density function(cdf) of $t$-distribution, define as in https://en.wikipedia.org/wiki/Student%27s_t-distribution#Cumulative_distribution_function I derive it by ...
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Studying extreme value r.v. $X=\max_i (c_i+X_i)$ where $c_i$ are constants and $X_i$ are i.i.d. r.v

Let $X_1,X_2,...,X_n$ be independently and identically distributed random variables according to a distribution $F$. There are constants: $c_1,c_2,...,c_n$. Define a new random variable $X=\max_i(X_i+...
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Test uniformity after adding some non-uniform data in python

I generate N1 numbers from uniform distribution using numpy python package from a certain interval e.g., ...
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Normal Distribution or Not

I have some net yield data (N=700) from a survey and I intend to carry out analysis to identify if there is any significant difference or relationships between the yield and the characteristics of the ...
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Questions the calculation of Z score and confidence intervals in biology

In biology, especially in some screening assays, people love to calculate a Z score between positive and negative controls. Essentially a distance calculation using the mean and sd to calculate how ...
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computing the expectation of a variable

I'm trying to run a simulation but before I need to compute the exact value of X, where: V ~ Bernoulli(0.5) X|V ~ Normal(1-2 V, 1) The question is : what is the expectation of X (E(X)) in the ...
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Shapiro Wilk test of normality

I ran my data under JASP and the p-value for the shapiro wilk test of normality was <.001. How do I interpret this?
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Return levels GEV distribution?

I am studying extreme value theory and I have a problem with the return level graph. I don't understand the convexity of the curves as a function of the sign of the shape parameter. In the book "...
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Visualization of $L(X)$

Let $L^2_+$ the set of all $2$-dimensional nonnegative random vectors $X = (X_1, X_2)^⊤$ with finite and positive marginal expectations, and let $Ψ^{(2)}$ the class of all measurable functions from $\...
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How to find expected value from cumulative distribution function?

Hello everyone, I'm currently doing research based on the model in Competitive fit-revelation sampling and mixed pricing strategy (Wu &Deng, 2021). And I don't understand how they can conclude the ...
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2 votes
2 answers
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Universal approximation of Gaussians

Can gaussian kernels reproduce non continuous L2 integrable functions? ( Do non continuous L2 integrable functions lie in the RKHS constructed by a Gaussian Kernel?) Edit: I think my question is being ...
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Winning Bid probability [duplicate]

I had asked this question earlier, but I guess I was too late to respond to the questions. Hence reposting. Top 10 win probabilities I have a task at my hand to create some kind of model that can help ...
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7 votes
1 answer
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Sample uniformly from unit square conditioned on sum and product

Consider the following conditional distributions: \begin{align} X, Y \stackrel{\text{iid}}{\sim} U(0, 1) &\mid X + Y = a & a \in [0, 2] \\ X, Y \stackrel{\text{iid}}{\sim} U(0, 1) &\mid X ...
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Question on solution of Casella and Berger Exercise 9.10: Showing that $Q(t,\theta)$ is a pivot

My question concerns Exercise 9.10 of Statistical Inference by Casella and Berger: On page 428 the authors say In general, suppose the pdf of a statistic $T$, $f(t|\theta)$, can be expressed in the ...
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How general is this property about correlation and the sum of two normal RVs?

(Cross-posted from math stack exchange as I didn't get any responses there) Given a random vector $(X_1,X_2)$ that is jointly normal with means / sd's $\mu_1,\mu_2, \sigma_1,\sigma_2$ and correlation ...
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Statistical methods for different parts of a distribution

I have a distribution of discrete values (grades, from low to high) where I see that the certain groups have a higher frequency of observations in the higher ranges and less in the lower ranges than ...
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Estimate parameters of an unknown negative binomial distribution based on known distribution

The PDF of a known NBD given in Equation (1). The parameter a and r are function of $μ$ = sample mean, and $s^2$ = sample variance, as given in Equation (2) and (3) respectively. $r$ = number of ...
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Finding an optimal split of a sample, based on distance of sub-sample's empirical distributions

Consider a sample $X$, consisting of $n$ observations that are independent (but not necessarily identically distributed), as well as two splits $S_1 := (X_{1,\dots,i}, X_{i+1, \dots, n})$ and $S_2 := (...
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1 vote
1 answer
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Get probability distribution function from density function and calculate the cumulative value [duplicate]

For the given density function, how to find its distribution function and how to calculate the value of the distribution function? Density function: $$f(x) = \frac{1}{\Gamma(\frac{n}{2})}x^{\frac{n}{2}...
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Find the probability of which sample comes from a "higher" distribution based on random sample from two distributions

I am trying to find the answer for the following question. I have two distributions of numbers between 1 and 8, as illustrated here: I draw random samples from each distribution, which we can call &...
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Top 10 win probabilities

I am a newbie here, hoping to find the right response for my question. I have a task at my hand to create some kind of model that can help me to determine what is the chance of winning a certain ...
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1 vote
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Methods to handle an arbitrary peak within a distribution

I had someone come to me for advice on how to deal with their weird data. The predictor variable is categorical (experimental condition), and the outcome variable is continuous (amount of money in a ...
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1 answer
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Infinitesimal Robustness, influence function of $T$ at $F$

This text is taken from Introduction to robust estimation and hypothesis testing. Wilcox R. First I will write down the description that leads to definition of relative influence on $T(F)$ and then I ...
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