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Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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Distribution of difference of two random variables with chi-squared distribution

Supose that we have two random variables $X \sim \chi_k^2$ and $Y \sim \chi_k^2$, with the same degrees of freedom. A chi-squared distribution cannot have zero degrees of freedom, so what would be ...
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Cumulative distribution function of a squared laplace random variable

I am trying to calculate $F_Y(x)$ (CDF) of $Y=X^2$ where $X$ is a random variable of Laplace Distribution $f_X(x) = \frac{1}{2}e^{-|x|}$ (let's take a simple case when parameters $\mu=0$ and $b=1$). ...
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How to Calculate Percent of Sample mean falls within Population Mean?

Following screenshot is from Udacity Statistics Tutorial. Klout score which is Social Media popularity score is distributed as shown in picture. Population mean is 37.72 and Standard Deviation is ...
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25 views

A Normal random variable divided by a Chi-squared random variable

I am looking to find the pdf of the ratio $Z = \frac{X}{Y}$, where $X \sim$ Gaussian and $Y \sim$ Chi-squared are indepedent random variables. A reference is good enough if you do not want to write ...
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1answer
22 views

How Probability distribution relates to neural networks?

The concept of random variables and probability distributions are confusing in the context of neural networks. In a neural network, which is the random variable and what is the probability ...
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35 views

Unusual distribution after 10000 iterations of multinomial experiment

I have a ground truth list (gt) which has x observations (time series), each observation can be 1 out of 3 possible classes (b,w,n). I have a classifier C that outputs x observations and I compute the ...
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1answer
47 views

Why data points are thought as random variable

I'am currently following a basic statistics course and a machine learning course. I try to understand what is covariance. In general, books define covariance as follows: Covariance is a measure of how ...
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10 views

Kolmogorov-Smirnov test on data from same subjects

My ultimate aim is to determine if drug effects are different from control in a given experiment.Can I perform Kolmogorov-Smirnov test on data obtained from a repeated measures design from a single ...
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Moments of the Generalized Dirichlet distribution

I have been trying to solve the following integral. $ \int\theta_j \sum_{k=1}^K \theta_k \beta_{k,w} \prod_{k=1}^K \frac{\Gamma(\alpha_k + \beta_k)}{\Gamma(\alpha_k)\Gamma(\beta_k)} \theta_k^{\...
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General techniques for coupling a set of random variables with mutual dependence

Disclaimer: the usage of coupling is in the title is not of the usual definition in probability theory. Suppose I have a set of random variables $\{X_1, X_2, \dots, X_n\}$, indexed by time $t$, and ...
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1answer
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Estimate a probability distribution of target values using features

In my particular problem, I have $$t \in \{1,...N\}$$ time periods, and feature vectors $$x_t \in R^m $$ which I hypothesize predict something about the probability distribution that the targets $...
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Rewrite a constraint on the probability distribution using the cumulative distribution function

Consider a probability distribution $P: \mathbb{R}^3\rightarrow [0,1]$ and assume $$ (\diamond) \text{ }\int_{(x_1,x_2,x_3)\in \mathbb{R}^3 \text{ s.t. } x_3= x_1-x_2} dP=1 $$ Questions: Is there a ...
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Statistics Question regarding probability distribution function [on hold]

Suppose that two random variables, X and Y have a continuous joint distribution with the following joint probability density function (pdf): 𝑓(𝑥, 𝑦) = 3/8*(2𝑥 + 5𝑦^2), zero < 𝑥 < 1; zero &...
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Hypergeometric Confidence Intervals

I need to create a confidence interval for the mean and variance of hypergeometric distribution of random numbers in R. Does anybody know how I can do this?
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How to derive joint CDF Gumbel distribution

If you have 3 random variables: $X$, $Y$, and $Z$ and they have independent Gumbel distribution. $A$, $B$ and $C$ are three discrete random variables that are functions of $X$, $Y$, and $Z$ as per the ...
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1answer
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How do you find shape parameters of a Johnson distribution from skew and kurtosis?

I'm trying to generate random numbers from a distribution with given moments, in my field the Johnson family of distributions is typically used but I can't find anywhere how to set it's shape ...
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Is there a distribution such that a sum of its samples creates the uniform distribution? [duplicate]

Say I sample $N$ numbers from a mystery discrete distribution and add them all together. My goal is for the sum of these random variables to be uniformly distributed from $[a,b]$ and thus the ...
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1answer
48 views

Distribution fitted well but kolmogorov -smirnov test not showing right results? Am I doing it right?

I want to do Kolmogorov -Smirnov test to see whether my data follows a particular distribution or not? When I fitted my data to lognormal distribution it fitted well. But When I am doing ks test, it ...
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1answer
34 views

Multivariate t approach multivariate normal

It is well known that in univariate case, as we increase the degree of freedom in a t distribution, it will limit to normal distribution. Does the result hold true for the multivariate case as well?
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How to distribute (depending on a probability) a categorical variable based on a conditional variable? [closed]

I have spent quite a lot of time on this but haven't found a way to do this in R. I have to distribute Category 2 based on what values are present in Category 1. For instance, in our example 2/3rd ...
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1answer
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What is the expectation of $\log(1+aX)$ where $X$ has an $F$ distribution?

The random variable $Y$ is defined by $Y=\log(1+aX)$ where $X$ has an $F(m,m)$ distribution and $a$ is a non-negative constant. What is the expectation of $Y$?
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1answer
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Proving a remainder term converges to 0 in probability

So we have these definitions: σ̂^2_1= (1/n)∑(Xi−μ)^2 σ̂^2_2= (1/n)∑(Xi−Xbar)^2 I have shown that n^0.5(σ̂^2_2−σ^2)= n^0.5(σ̂^2_1−σ^2)- n^0.5(Xbar-μ)^2 I am trying to show that the remainder term ...
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Expanding known distribution over time with an algorithm

I'm currently struggling with the following problem: An event reapeaditly occurs with an average interval of $m$ months for $n$ times (assume $m = 13.67$ months and $n = 3$ for the following example). ...
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Modelling my own cost function for skewed distributions

A while ago I worked in a model used for fraud detection, as that involved a very asymmetric distribution, I used F1-score as my loss function. However, I cannot stop thinking about it, and I feel F1-...
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1answer
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Finding joint support of $(XY,X/Y)$ where $(X,Y)$ has joint pdf $1/x^2y^2$ for $x,y\ge1$

$X$ and $Y$ are random variables with joint pdf $$\frac{1}{x^2y^2}\qquad,\, x\ge1,y\ge1$$ Set $$U=XY, V=X/Y$$ Explain why the joint range of $U$ and $V$ is given by: $$\{(u,v):v\in(0,1),u\ge1/v\} \...
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1answer
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Probability after getting distribution from marginal distributions

This is a problem on Freund's Mathematical statistics book on page 101 If the independent random variables $X$ and $Y$ have the marginal densities $f(x) = \begin{cases} 1/2 \space \text{for} \space ...
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1answer
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Can we conclude, with the strong law of large numbers, that $n$ random variables are independent? [closed]

Suppose we have a sequence of identically distributed random variables $X_1, \ldots, X_n$, and that we know $(X_1 + \ldots + X_n)/n$ converges almost surely to $\mu = E[X]$ as $n$ approaches infinity. ...
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KS test and QQ-plot with catagorized data. KS statistics a reasonable meassure for deviation from desired distribution

I am completely new in this Blog (in any Blog at all) and additionally I am not experienced in R and sorry for the english. However, I want to compare distributions from several samples (field plots ...
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Is there a name for the distribution whose PDF is -ln(x) on its support [0, 1)?

If so, what is its name? If not, how/where can information about it be found?
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1answer
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Confused about type of analysis between categorical variable (degree of severity of accident) and continuous variable (number of car passengers)

Good afternoon, As indicated in the title, for a project I am interested in finding any (if at all) relation between the number of passengers in a car and the severity of an accident. I am using JMP ...
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1answer
249 views

Does the 2-sample KS test work? If so, why is it so unintuitive?

Here is an example. I have two data sets as shown below as histograms: I want to examine whether these data sets are drawn from the same underlying distribution. To do this, I am using the 2-sample ...
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K-Nearest-Neighbor classification with only distance/similarity matrices, is it possible?

I want to classify histograms/distributions using K-Nearest-Neighbor. I can measure distances/dissimilarities between the distributions (using euclidean distance, kullback-leibler divergence...), thus ...
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Calculating statistics on transformations of many random variables

I have a model that uses $n$ independent random variables $X_1,..., X_n$. I know the density function of each random variable. I would like to calculate statistics such as $E(\sqrt{X_1+...+X_n})$ or $...
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Advice on choosing the most appropriate significance test (for some simple data)? [closed]

I'm asking for advice on how to pick right tests and mainly how to justify them. For the sake of data, I made a plot which shows a continuous approximation of the number of suicide-thought :| per ...
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How to measure the validity of the confidence intervals? [closed]

I'm trying to decode a problem and would need some guidance to do it. Usually I have a function which outputs the confidence intervals (min and max) of a distribution, but how do I validate if the ...
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1answer
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Looking for a statistical test which will test whether the “spread” from a line is statistically significant

Below are two examples of two x-y data pairs: The top example clearly has more "spread" from any sort of linear trend between x and y. The data points are clustered together but there are some ...
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1answer
21 views

Gradient-Descent for Parameter search

I have a rather straight-forward algorithm for finding the maximum-likelihood parameter of a probability distribution using sub-sampling. I'm fairly confident this algorithm is not novel and was ...
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A distribution that describes observing fragments of a multiple items of differing lengths.

Suppose I have some different colored sticks with a unique length for each color, I then cut the sticks in to 1cm chunks. Is there a distribution that describes the probability of me observing x ...
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From trivariate cdf to the distribution of differences of random variables

Consider a trivariate cumulative distribution function (cdf) $G$. Is there a collection of necessary conditions on $G$ ensuring that $$ \exists \text{ a random vector $(X_1,X_2)$ such that $(X_1, ...
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1answer
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calculation of paramters needed for joint probability distribution?

Please correct me if I'm wrong. From my understanding, the number of entries in the above image is 7 because you need to calculate 7 and the 8th one can be done by 1-p. But I can't understand how ...
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0answers
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How to measure the “variability” across a set of many (>>2) probability distributions?

Given a set of many of discrete probability distributions, is there a way I can efficiently calculate a metric that quantifies how different the entire set of these probability distributions are to ...
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3answers
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Why does the Kolmogorov-Smirnov test work?

In reading about the 2-sample KS test, I understand exactly what it is doing but I don't understand why it works. In other words, I can follow all the steps to compute the empirical distribution ...
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0answers
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Significance test in trimodal distribution [duplicate]

I have a trimodal null distribution that looks like this: In order to calculate the significance we just check how extreme the values are respect to our null distribution. We do it this way: ...
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1answer
38 views

What a tests do I use to show that two parts of observations come from a two-dimensional uniform distribution?

I have a two-dimensional uniform data. I have splitted the data on two parts with a ...
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2answers
119 views

Deriving an expression for a confidence interval for σ^2 using the asymptotic distribution of √n(σ̂^2−σ^2)

We have We have $X1,…,Xn i.i.d N(μ,σ^2) $where $μ$ is known and $σ^2$ isn't known. $σ̂^2=(\frac{1}{n})∑(X_i−μ)^2$. First of all what I did, I derived an equitailed 95% confidence interval for $σ^2$....
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What is the proper way to model reading & response times

I plan to measure the time it takes users to read and respond to a series of similar multiple-choice questions. Before starting the experiment, I'd like to think of a proper distribution that can be ...
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Exponential Family Representation: Dumb question on scale parameter and whether it went to Hawaii

So going over the Hastie Tibshirani paper on GAM - it points to equation 11 as the exponential family density - but with two parameters - theta (natural parameter) and phi (scale). https://...
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Distribution (newsagent example) [duplicate]

I know that the given distribution is a discrete uniform distribution to model the number of customers. How do I link this to the two possible events (given in the hint) to get the expected value of ...
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1answer
131 views

Proving Asymptotic distribution of $\sqrt n( \widehat\sigma^2 -\sigma^2)$

I am looking at trying to derive an expression for the asymptotic distribution. We have $X_1,\ldots, X_n$ i.i.d $N(\mu, σ^2)$. So we have defined $\hat \sigma^2 = \frac 1n \sum_{i=1}^n(X_i-\mu)^2$. (...
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Is there a method that's better than clustermaps for clustering samples which have a n of variables but only two possible outcomes?

For each variable there are two possible outcomes 1 or 0 (True or False): It shows the appearance of particular species on different localities. Mine goal is to cluster/classify similar localities ...