Questions tagged [distributions]
A distribution is a mathematical description of probabilities or frequencies.
7,922
questions
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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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27 views
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 $...
4
votes
1answer
54 views
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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0answers
57 views
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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0answers
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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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0answers
28 views
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 $...
1
vote
1answer
19 views
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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86 views
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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0answers
20 views
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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0answers
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 ...
1
vote
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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0answers
9 views
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 ...
1
vote
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 ...
2
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1answer
42 views
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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0answers
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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0answers
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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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129 views
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
31 views
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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27 views
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(\...
1
vote
1answer
16 views
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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0answers
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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0answers
17 views
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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0answers
11 views
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 ...
3
votes
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$, ...
3
votes
2answers
81 views
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
22 views
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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0answers
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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0answers
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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 ...
4
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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 ...
2
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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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0answers
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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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0answers
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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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0answers
11 views
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
18 views
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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0answers
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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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0answers
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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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0answers
31 views
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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0answers
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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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0answers
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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 ...
1
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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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24 views
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 ...
