Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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Variance of a sequence of bernouli(p) trials where p is drawn from uniform distribution [0,1]

A number p is drawn from the interval [0,1] according to the uniform distribution, and then a sequence of independent Bernoulli trials is performed, each with success probability p. What is the ...
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How discriminator output in GAN is a probability distribution?

Recently I asked a question about GAN,What is the intuition behind the expected value in orginal GAN papers objective function? , In there I came to know that the discriminator output is viewed as a ...
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1answer
32 views

What is the name of $D(F,G)=\int(F(x)-G(x))^2dF(x)$?

Is there a statistical distance between two 1-dim distribution F and G that $D(F,G)=\int(F(x)-G(x))^2dF(x)$? Or to symmetrize it, take $D^s(F,G)=\int(F(x)-G(x))^2dF(x)+dG(x)$ If not, why? (What are ...
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Forward or Reverse KL Divergence - based on available sampling

Let us say our KL Divergence equations are as follows: $$ \text{Forward KL Div.} = D_{KL}[P_X || P_{F(Z)}] $$ $$ \text{Reverse KL Div.} = D_{KL}[P_{F(Z)} || P_X] $$ Let's say we have 2 cases, one ...
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1answer
43 views

Need help with understanding random variables/the data generating distribution

Lets say we want to predict a persons weight using their height and gender. We always assume there is a data generating distribution $P_{X×Y}$, and all output and input pairs are generated i.i.d from $...
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1answer
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Hit probability of a motif in DNA sequence

I am trying to figure out expected number the occurrences of a DNA motif in a sequence. Let me first show a small example for the base case: Assume we have a DNA sequence 'TACCTG', where in every ...
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What's the joint conditional distribution of 2 random variables?

What's the joint conditional distribution of 2 random variables? Please provide a reference for the definition. (I forgot if any 2 continuous random variables necessarily have a well-defined joint pdf....
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Compute probability of an event by integrating over continuous random variable

Let $B$ be a continuous random variable. Let $K$ be an event. Am I right to think it does not necessarily make sense to say '$P(K)=\int_{b \in \mathbb R}P(K|B=b)f_B(b)$'? My guess: Well based on ...
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Show that if $Y \sim X + \delta(X)$, then $G(\cdot)= F(\cdot-\Delta)$ implies $\delta(x) \equiv \Delta$

Problem is from Bickel - Mathematical Statistics, which I am working through on my own. Suppose that $Y \sim X + \delta(X)$ where $X\sim N(\mu,\sigma^2)$ and $\delta$ is continuous. Suppose also that $...
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1answer
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Does Kernel density estimation normalise the distributions?

I am analysing polymorphisms distribution data from Next Generation Sequencing data using Kernel density estimation (KDE). However I would like to know if this method permit an unbiased comparisons, i....
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Conditioning on probability zero: Can we say $P(B \le A|B=b) = P(b \le A|B=b)$?

Let $A,B$ be continuous random variables. Let $E,G,H$ be events. Let $t \in image(B)$. (I forgot if any 2 continuous random variables necessarily have a well-defined joint pdf. If not, then assume ...
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Is there a name for this generalisation of the exponential distribution

Is there a name for the following: $$ f(x) = \lambda(x) e^{\int_0^x -\lambda(t) dt} $$ which is similar to an exponential distribution. If $f(x)$ is a polynomial, would this be classed as a gamma ...
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26 views

what is the standard deviation of the geometric mean sample distribution?

I wrote a python script to take a population distribution of a random variable in the interval (0,1) to be uniform and make 2 sample distribution: The fist is the distribution of the arithmetic mean ...
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1answer
43 views

Bias correction for regression with t-distributed error

I have a GAM /regression model which is originally defined as: log10(Y)~s(log10(X1))+s(log10(X2))+s(log10(X3)) #using R mgcv The response needs to be back ...
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Does different data distribution of training and testing data cause overfitting?

Let's assume that I'm developing a classification model for the product of my company but there's a problem. The problem is the data from my company is not enough to develop the model since my company ...
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1answer
29 views

Evaluation of Propensity Score Matching: quantify bias of variation in sample distribution

I've completed propensity score matching of a treatment and control group across a number of covariates. On two categorical covariates we require an exact match, for example, Gender and Eye Color, and ...
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1answer
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Distribution of Difference between Functions

I am attempting to find the distribution of the difference between two functions. In the images below I have one function in green, and one function in black defined by the red points. When I ...
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22 views

Sufficient statistic for a given distribution from exponential form

Given a particular form, i can verify whether it is sufficient statistic or not using $\frac{p_\theta(x_1,x_2...x_n)}{p_\theta(T(x_1,x_2...x_n))}$ is independendent of $\theta$ then i can say $T(\bar ...
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How to analyze usefulness of imputation output in R

I am working with a dataset with 3,500 observations and includes a Body Mass Index variable. There are around 300 NA values for the BMI variable which I have imputed using multiple imputation. ...
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Checking the distribution of input variables with missing data imputed in exploratory data analysis

Suppose there are missing data in input variables and the missing rates are relatively high, we use some certain value to impute the missing info. When we check the distribution for input variables in ...
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What distribution has the p.d.f. of form $\frac{e^{-b \cdot x} \cdot x^c}{1 + e^{-k \cdot (x+a)}}$ for $x >0$ and parameters $b > 0, k > 0, c > 0$?

In my work I met the p.d.f. of form $$p(x) \propto \frac{e^{-b \cdot x} \cdot x^c}{1 + e^{-k \cdot (x+a)}}$$ with support $x \in [0, +\infty)$ and positive parameters $b > 0, k > 0, c > 0$. ...
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1answer
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Does the “3 sigma” rule and the Tchebyshev inequality use the “by-definition” standard deviation and mean or the distribution-specific ones?

When we speak about the 3xSD rule for the normal distribution, or the Tschebyshev inequality, saying how many data are covered by the appropriate multiplicity of SDs, do we mean the square root of the ...
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Specifying the Sampling Distribution of the Sample Means When the Population Standard Deviation is Unknown

For a random variable: $$X\sim N(\mu,\sigma)$$ If the population standard deviation is known, then we know that the sampling distribution of the sample means is described as: $$\bar{X} \sim N\Biggl(\...
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1answer
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How to calculate number of cycles in twenty year from a daily distribution

I am asked to determine a distribution of the number of cycles a component is expected to do during its entire lifetime. What is do have now is a distribution of cycles the component makes per day. I ...
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24 views

Maximum information in dichotomous variables?

I have a number of dichotomous variables from a number of study participants, and I want to do some exploratory factor analyses and other things. However, my sample is too small to include all the ...
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Determine distribution for GLM

I would like to determine the distribution for the following data in order to explain it with a generalized linear model and further parameters: A K-S test for Poisson distribution has already been ...
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Determining if two groups of uneven size have the same distribution?

I have two groups, an experimental of ~100 and a control group ~1000. I want to pick a selection of 100 from the control group at random to be the proper control for the experimental group. Before I ...
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1answer
131 views

3 uniform points on a circle

Suppose 3 (distinct) points are uniformly and independently distributed on a circle of unit length (smaller than a unit circle!). This is really circle and not disc. Call one of these points $B$. Let $...
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1answer
35 views

How to interpret KS statistic and p-value form scipy.ks_2samp?

I just performed a KS 2 sample test on my distributions, and I obtained the following results: CASE 1: statistic=0.06956521739130435, pvalue=0.9451291140844246; CASE 2: statistic=0.07692307692307693, ...
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Checking for consistency in pairs or triplets of distributions?

Background I am involved in a very large project studying plants. We have millions of plants, and for each plant, two or three different research groups are going to measure three properties $X$, $Y$, ...
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2answers
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CDF of $M(X,Y)$?

Let $X,Y \sim^{\text{iid}} Unif(0,1)$. Let $M = M(X,Y) = \min\{X,Y,1-Y,1-X,|X-Y|,1-|X-Y|\}$. Supposedly $image(M) \subseteq (0,\frac13)$ and distribution of $M$ is $F_M(m)= (3m(2-3m))1_{(0,\frac13)}(m)...
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1answer
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Forecasting probability distribution for year-ahead resolution

Background I am currently working on a problem to study the dynamics of aggregate losses in some state-owned companies in my country. I was successful in gathering data of losses across 34 such ...
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26 views

How to interpret this Residuals vs. Fitted Plot [duplicate]

How to Check the non-constant variance using this ’residuals vs fitted values’ plot?
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Distribution of a rate in a subsample after permutation of the total dataset

I need to estimate the distribution of a rate in a subsample after permutation of the total dataset. I was wondering whether it's sufficient to use the binomial distribution with the global rate ...
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1answer
35 views

What is the distribution of time's to ruin in the gambler's ruin problem (random walk)?

In a gambler's ruin problem, where the gambler starts with a fixed amount of wealth. What is the distribution of times to ruin. That is, if each bet has a fixed payout. As I understand it, this is a ...
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1answer
37 views

When is the ratio of two normals approximately normal?

Suppose that $X \sim N(\mu_1,\sigma_1)$ and $Y \sim N(\mu_2,\sigma_2)$ are two independent normal random variables. Define $Z = X/Y$. I noticed that there are some cases where the distribution of $Z$ ...
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Limit distribution for a linear discrete-time stochastic process: limit of the sum of linearly transformed uniform distributions

I have posted this in math stack exchange, but I figured maybe this is a better forum for this kind of question. Suppose we have a stochastic linear process: $$x_{k+1} = Ax_{k} + Bw_{k} \qquad \text{...
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Proof that ratio of normal distributions is Cauchy [duplicate]

I recall that the ratio of 2 independent standard normal random variables follows a Cauchy distribution, but I don't recall if there is a proof for this.
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Parametrization by loc/scale or mean/std?

I need a clarification on are there any differences in parametrizing probability distributions by location and scale vs. mean and standard deviation. For example, Python's SciPy does standardization ...
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Comparing Feature's Distribution in Logistic Regression

I was running Logistic Regression on some data. After some RFE, correlation testing and so on, I was expecting some features to play a determining factor. However, seeing at the coefficients given ...
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1answer
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Nearest neighbour distribution empirical discrepancy in high dimensions

I'm attempting to use the nearest neighbour distribution to understand the separation between uniformly distributed points in a high-dimensional space. I find that there is discrepancy between ...
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Bayes estimate of upper limit of uniform distribution with exponential prior

Let $X_{1}, . . . , X_{n} > 0$ be a random sample from $U(0, \theta)$. Suppose $\theta$ has the prior $\pi(θ) = e^{-\theta} ; \theta > 0$. Find the Bayes estimate of $\frac{1}{\theta}$ with ...
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How to perform a $\chi^2$ test when the expected value is zero

I have performed an experiment and obtained the following data: The black dots are the recorded values at a given time and the red line is a curve of best fit. Now I want to perform a chi-squared ...
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Analyzing grade distribution across 3 types of courses

I am trying to look at the final grade distribution in 3 types of classroom; in-person, hybrid, and online. These are the same 2 courses that were each taught in the 3 different formats. What is the ...
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Finding Confidence Interval for Lower Bounded Truncated Normal Distribution

I am working on finding a confidence interval for data that follows a lower bounded truncated normal distribution (lbtnd) bounded from 0 to $\infty$. I am having difficulty completely understanding ...
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How to deal with a PDF that takes CDF data as input? (Metalog distributions)

I am trying to utilise the Metalog distribution in a Machine Learning project. For this project, I need to be able to obtain likelihoods using the PDF of the distribution. https://en.wikipedia.org/...
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Determining if method captures trends in data

I am working in a problem where a method was developed to capture a set of features $X$ in an image for two conditions - control $c$ and disease $d$. To do this, we train a machine-learning model that ...
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1answer
53 views

Expectation of balls in a jar

Trying to solve this question: A jar contains 10 balls numbered 1,2,3,...,10. We draw 15 balls from the jar, one after the other, with replacement. Let N denote the number of distinct numbers drawn. ...
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31 views

Which distribution integrates well with a sum of random variables raised to a power?

Imagine I have three random variables $x$, $y$ and $z$, and I have a function of these three random variables $f(x,y,z;a,b,c)=(xa+yb+zc)^{\alpha}$, for $\alpha>0$. The random variables are non-...
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Monte Carlo sampling ( accept/reject) for geographic dataset

I have a dataset consisting of latitude and longitude and I'm confused on which approach to use to determine the distribution of points so I can apply the monte Carlo accept/reject for sampling. this ...