Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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the mean and standard deviation aren't the same as those of the input data i provided after sampling

I have a log-normal mean and a standard deviation. after i converted them to the underlying normal distribution's parameters mu and sigma, I sampled from the log-normal distribution however when i ...
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Joint gaussian conditional on its sum greater than a value

Let us consider $X\sim\mathcal{N}(\mu,\Sigma)$ being a $d$-dimensional multivariate Gaussian random variable. I know that it is possible to calculate the distribution of $X|S=s$, where $S$ is the sum ...
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t- test for non normally distributed sample

I am doing a statistical test (analysis) for the following case: As part of a product aimed at improving the quality and speed of code writing for developers, we have implemented a new feature that ...
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Simulate Arrival Time for Scheduled Appointments

I'm trying to build a simulation of a healthcare center. The appointment times are scheduled every 30 minutes. However, some patients arrive later than their appointment and some might arrive early. ...
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transformation of a kernel density estimate to uniform distribution

I am interested in estimating the expected value of a function, $f(x)$ with respect to a probability density function, $P(x)$. I am exploring a method that requires I change variables from the ...
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Obtaining a tractable expression for P(X<2Y)

The joint probability of the bivariate random vector $(X,Y)$ is given by $f_{X, Y}(x,y) = \frac{1}{2\pi}e^{-\sqrt{x^2+y^2}}$. Compute $P(X<2Y)$. My attempt: I made the obvious transformation to ...
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The reason behind fixing the IQR coefficient value in Interquartile Range method

To find the outliers, one common approach is to use the Interquartile Range method, especially when you know your data does not follow Gaussian distribution. ...
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Detect and remove outliers from unknown distribution

I have completed a range of steady-state CFD simulations on building roofs. A contour map of the resulting variable is displayed in the Figure below with the corresponding values on the left side. ...
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Sorting samples from distribution before calculating the distance [duplicate]

I have to evaluate different methods for distribution fitting. So, given an sample set A I get a fitting distribution B or I might get another sample set C that is much like A. Now I need to do a ...
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Distribution of half-life from radioactive decay

Suppose that $X$ measures the half-life of a radioactive element, with decay rate $\lambda$ (per unit of time). Starting from a population of $N$ particle, I believe you can model the number of ...
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Understanding the relationship between a 'sampling distribution of a statistic' and a 'population distribution' in the context of a 'hypothesis test'

I would like to confirm that I am understanding the relationship between a sampling distribution of a statistic (an example of a 'statistic' would be a sample mean $\bar{x}$) and a population ...
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Distribution of a function of Normal and Uniform random variables

consider the distribution of $$Y=\sqrt{(A \sum_{i=1}^{k}\cos \theta_i+\Re{\{ N \}})^{2}+(A \sum_{i=1}^{k}\sin \theta_i+\Im\{N\})^{2}}$$ where $A>0$, $N \sim \mathcal{CN}(0,2\sigma^2)$ and $\Theta_i ...
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Estimating stationarity of evolving distribution

Consider a probability distribution $P_t$ at time $t \in \mathbb{N}$ which evolves in time, but is assumed to become stationary at some unknown time $t_0$, i.e $P_{t_0+1} = P_{t_0}$ To make matters ...
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Sum of values with different probabilities

Suppose I have the following linear expression: $S = x_1 + x_2+ \dots + x_n$, in which each $x_i$ can only assume the following values: -2, -1, 0, 1, 2 whose probabilities are 0.1, 0.2, 0.2, 0.25, 0....
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Histogram vs Bar Graph [closed]

I have data for number of patients used to collect the data in a hospital for each day. I want to display them as a graphical representation of number of patients used to collect the data for each ...
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How does detailed balance relate to (conditional) expectation?

Let $(\mathsf{X}, \mathcal{X})$ be a measurable space and $\pi$ be a probability distribution on it. Let $\mathrm{K}:\mathsf{X}\times\mathcal{X}\to[0, 1]$ be a Markov kernel. We say that $\mathrm{K}$ ...
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Kolmogorov-Smirnov inconsistency with increasing dataset sizes

I want to compare two arbitrary distributions and know that Kolmogorov-Smirnov is one such test that can handle any non-normal distribution. I noticed however, that p-values returned by the KS test ...
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Is it true that $\mathbb{E}_{X}[f(X)] = \mathbb{E}_{X, Y}[f(X)]$?

Suppose $p(x, y)$ has $x$-marginal $p(x)$ and that $f(x)$ is a well-behaved function. Is it always true that the expectation with respect to the joint distribution is equal to the expectation with ...
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Population statistics

Let’s say one has a small random size of 100 people, and the distribution of this small population is 60% male and 40% female. Now, say, there are 1,000,000 people that live in this region in total. ...
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Evaluate the goodness of a fitted continous distribution in the case of ties (Used later in DES simulations)

I have continous data which shows the time in minutes, a process step takes. Unfortunatelly, it is not possible to get data without rounding so I have many ties. As I thought, a random process should ...
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Salary of a group of people is continuous or discrete

I have salary data of 3000 employees ranging from 3000 - 10000 dollars. Based on my understanding:(https://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGContinuousDiscrete.html) Continuous data is ...
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How to validate the decomposed distributions?

I am fitting distributions for the time spent for three processes (i.e., pick up tools, walk to destination, install) in a system that I am trying to simulate where the original data for these ...
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How does posterior predictive mean depend on parameters of the likelihood and prior distribution?

I have come across a problem in my research which deals with the mean of the posterior predictive distribution, i.e. $$p(x'|x)=\int d\theta p(x'|\theta)p(\theta|x)$$ where $x$ is an observed sample ...
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Finding an optimal distribution to fit to a highly skewed data vector (DV with missing values) in R

I hope that this question has not already been asked. I am analyzing data in R (and am a novice). I have a highly skewed data vector in a dataframe with missing values that I hope to set as the ...
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Probability distribution for stay duration in a location ( e.g., a restaurant)

I have a user mobility traces dataset where each user trace is defined as a sequence of GPS coordinates and the type of location (e.g., coffee shop, office, restaurant) as depicted below: ...
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Expected value of values in the interval

There is a normal distribution with EV 100 and SD 20. I take a random number, say 63 How to calculate EV of all values that follow the original distribution but excluding those bigger than 63? In ...
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What metric is best fitted for comparing encodings?

I am trying to compare two distributions, that each correspond to different numerical encodings, e.g. compare fp32 encodings to various other encodings on a same set of values. However I do not know ...
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Empirical error in Kullback-Leibler KL divergence estimation

In computing the Kullback-Leibler KL divergence $D(P\|Q)$ from an empirical data, it may happen that $Q(x)=0<P(x)$ at some sample point $x$ due to data error and $D(P\|Q)=\infty$. What are some ...
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Deriving marginal distribution from a funky joint exponential-like distribution

Let $X$ and $Y$ be two random variables such that $$P(X > x, Y > y) = \exp\lbrack-(\lambda_1 x + \lambda_2 y + \lambda_{12} \max(x, y))\rbrack$$ What is the marginal distribution of $X?$ Source ...
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How to fix intersection of cluster distributions in R

I need help with a clustering task I'm doing. The essence of the problem, there is data on vegetation indices. Simple example for R ...
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Connection between forms for Generalized Pareto Distribution

On Wikipedia (https://en.wikipedia.org/wiki/Pareto_distribution#Pareto_types_I–IV) one can find the relation between the different types of Pareto Distribution and the Generalized Pareto Distribution (...
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What is $F_{k,\infty}$, i.e., $F$ distribution when the second degree of freedom approaches infinity?

What is $F_{k,\infty}$, i.e., $F$ distribution when the second degree of freedom approaches infinity? I'm wondering if there is a known distribution(such as $\chi_k^2$) that it converges to.
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Sample of Runners - Can the Group Run 2.5 Miles in 20 mins?

I have a dataset where there are 6 runners. Each runner runs as far as they can for 20 mins, and a watcher records their distance (to the nearest 0.1 miles) at certain times, precisely on the minute ...
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A\B test and sample size calculate for time data

I am doing a statistical test (analysis) for the following case: As part of a product aimed at improving the quality and speed of code writing for developers, we have implemented a new feature that ...
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Joint support of a bivariate random vector

I encountered a statistics problem that seems basic and I feel that I am being tripped by the graph of part c). Since part c) is the only part I feel I need help with, I will only mention what I got ...
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Proving $\sum_{i=1}^n(X_i-\overline X_n)^2-\sum_{i=1}^m(X_i-\overline X_m)^2 \sim \chi^2_{n-m}$

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d $N(0,1)$ random variables. For $2\le m<n$, let $S_m^2=\sum_{i=1}^m(X_i-\overline X_m)^2$ and $S_n^2=\sum_{i=1}^n(X_i-\overline X_n)^2$ where $\overline X_m=\...
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When is it favorable to not visualize univariate data using empirical CDFs?

A straightforward question: when is it "favorable" to interpret visualizations using, say, boxplots, histograms/density estimates, versus empirical cumulative density functions (ECDFs)? I've ...
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Why does "Hoeffding's bound greatly overestimates the probability of large deviations for distributions of small variance"?

I've read in a paper using Hoeffding's inequality to derive a bound on the probability of the difference of means of two samples being larger than a threshold that "Hoeffding's bound greatly ...
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Does the density $g(y) \propto (1-y^2)^{(n-3)/2} e^{\delta y} \quad\text{for}\quad |y| \leqslant 1$ have a name?

The following probability density function has a particularly simple form, and it was produced when deriving a confidence interval for $\frac{\mu}{\sigma^2}$ , $$g(y;\delta)=c_\delta(1-y^2)^{(n-3)/2}e^...
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Sampling Puzzle

There is a bag with N = 50 balls. Among which M = 10 balls are red, and N-M = 40 balls are blue. Further, say the red balls are numbered among themselves from 1 to 10, and the blue balls are numbered ...
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Probability of failure of Uniform Sampling [duplicate]

Say I have a bag with 10 numbered balls, and I pick one ball at each time step and then put it back in the bag. Since each ball is equally likely, therefore the current situation represents a uniform ...
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Why are my P-Values for the scipy kstest on a uniform distribution so high?

I'm using the Scipy implementation of the Kolmogorov–Smirnov test to check whether collections of random values are likely to have been drawn from a uniform distribution. From what I understand, the ...
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What do ggplot's stat_summary errorbars mean?

Say we have the following data: ...
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Synthasize data given mean, variance, skew, and kurtosis in python

I would like to generate synthetic data by specifying their mean, variance, skew, and kurtosis. However, I only know how to generate synthetic data with mean and var. Here is an example with mean and ...
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Likelihood and cross-entropy: continuous case

I think it's pretty clear to me that average log-likelihood is equivalent to negative cross-entropy for discrete distributions, as shown here: $$\frac{1}{N}\log\mathcal{L}(\theta) = \frac{1}{N}\log \...
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What is this type of data called?

An event occurs once per period, such as once per year. Time is measured in discrete units, such as days of the year. Let $A_y$ be the day in year $y$ on which this event occurs. However, we do not ...
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How to deal with limited sampling when analyzing a probability distribution

I constructed a probability histogram for my data and wish to construct a log-log plot to look at the distribution, but the sampling (I'm not quite sure how to phrase this) is very poor at the tail ...
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Simulating Horseshoe Distribution

Can someone check whether I correctly simulate the Horseshoe pdf? ...
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covariance of lognormal random variables

I am trying to find the variance of b*log(x+y) - log(x), where x and y are independent and identically distributed lognormal random variables, the range for log(x) and log(y) is negative infinity to ...
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Basic GLMM model fitting

I'm fairly new to fitting GLMMs but hoping I can get some advice. I have run an experiment where participants in an intervention and control group each perform a task (A) and then perform a similar ...
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