Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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How to define a skewed Gaussian distribution using mode and two points? [duplicate]

I want to define a Gaussian distribution function and plot it in python using the mode and two points parameter values instead of using the mean and standard deviation. For example, I have ...
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45 views

Hypergeometric distribution- problem with derivation

A random variable $K$ has hypergeometric distribution with parameters $N, m, n$, with probability mass function: $$ p_K(k)=\frac{\binom{m}{k}\binom{N-m}{n-k}}{\binom{N}{n}},\quad k\in\{\max(0,n+m-N),\...
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36 views

Obtaining marginal PDFs through change of variables

Given a random variable $\textbf{x} = (x_1, x_2, \ldots, x_D)$ with multiple dimensions and PDF $p_X(\textbf{x})$ and some invertible transformation $\textbf{y} = f(\textbf{x}) = (y_1, y_2, \ldots, ...
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Which financial time series have a PDF and/or a CDF?

Consider the following types of financial time series for a single publicly-listed stock: Price data Log returns Cumulative returns Each is computed from the item listed before it: log returns are ...
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1answer
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Nonparemetric tests: how to support the null hypothesis you claim to be testing

Let us assume that we have taken an unbalanced number of independent random samples from 5 different populations, which will be analogous to 5 different locations in this example. Each observation ...
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Find an expression for the cumulative distribution function of a random variable

Let $X$ have distribution $F$ and density function $f$ and let $A$ be a subset of the real line. Let $I_A(x)$ be the indicator function for $A$: $$I_A(x) = \begin{cases} 1 \hspace{5mm} x \in A\\ ...
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2answers
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Modeling Pseudorandom Number Generator Without Replacement

I vaguely recall a definition of a pseudorandom generator from cryptography. My rephrasing here: No adversary can (with non negligible probability) predict the next bit of a uniformly (pseudo)random ...
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Dealing with multiple fine-tuning steps

I have this Scene Text Recognition network that I'd like to fine-tune on a completely different domain (same task). I have just few examples of this target domain (let's call this dataset T). So, I've ...
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1answer
27 views

How to deal with zero inflated columns in dataset?

I have a dataset on which I am trying to fit a Linear Regression model. It has 4 independent variables. I am trying to predict my dependent variable using these four columns. However, 2 out of these 4 ...
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1answer
37 views

Is there a central limit theorem for random variables with a bounded interval? [duplicate]

Is there any theorem which states the asymptotic distribution for the sample mean when the samples are drawn from a random variable which has a bounded interval?
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Constrained optimization of joint Bernoulli density parametrized by a neural network

I want to learn a joint distribution on $n$ Bernoulli random variables conditioned on a random variable $b\sim D$ and parametrized by a neural network, $f$: $$p(A = (a_1, \ldots, a_n)|b) = f_{\Lambda}(...
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1answer
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Easy vs difficult distributions for sampling

Many sampling methods (e.g. rejection sampling) approach the sampling of a distribution $p$ as a problem of sampling from a different and somehow easier distribution $q$ and then correcting or ...
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How should I transform right-skewed percentage data with zeros? [closed]

I am dealing with a data set on student attendance, or more specifically, student absence rates. The data is right-skewed, and most values fall between 0 and .3. What kind of transformation can I do ...
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Estimate parameters in Matlab (multivariate t distribution)

Is there a built-in function in Matlab that estimates parameters of a multivariate t distribution (scale matrix, degrees of freedom). In other words, I have time series of returns for 10 stocks. ...
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1answer
65 views

How to fit a distribution onto a CDF data obtained from some analysis. [closed]

I have CDF data that is coming from a reliability analysis and I want to fit a distribution onto it so that I can perform further analysis. Note that I don't have the actual data set but only the CDF ...
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continuos uniform distribution pdf value at upper bound

What is the most formal (and coerent with probability theory) definition for the value of pdf(b) where b is the upper bound of the support of the continuos uniform distribution U(a,b) ? We can choice: ...
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Compare distributions using Maximum Mean Discrepancy (MMD)

I use MMD distance to run a permutation test and decide whether two sample distributions come from the same distribution or not. For the MMD, I use a gaussian kernel, the bandwidth of which I select ...
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1answer
37 views

Forecast confidence intervals from multiple realizations

I have a forecast which involves sampling a probability distribution and therefore each time I run the forecast there is some random variation between results. If I run the forecast many times, how do ...
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1answer
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Why is there a kurtosis condition for joint distributions to be elliptical?

I read that if x1, x2 are 2 random variables with different excess kurtosis, their joint distribution cant be elliptical. Is there an intuition or proof of that? It is not very clear to me. Edit- in ...
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1answer
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Intuition for expectation of discrete random variable that takes positive integers

If $X$ is a discrete random variable that takes values on the positive integers, it is true that $$E(X) = \sum_{k=1}^{\infty} P(X \ge k)\;.$$ I know how to prove this (by expressing the summand as a ...
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1answer
52 views

Compute 16 var(x)+32 var(y) for given bivariate CDF

\begin{equation} {F(x,y)} = \begin{cases} 0 & \text{if $x<0$ or $y<0 $} \\ \frac{1-e^{-x}}{4} & \text{if $x>0, 0 \leq y <1$} \\ 1-e^{-x}& \text{if $x \geq 0, y \...
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1answer
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Doubt in independence of 2 random variables

If 2 random variables are independent, then $f(x,y) = f(x)f(y)$. Is converse true? $F(x,y)=F(x)F(y)$. Is converse true? $E(x,y)=E(x)E(y)$. Is converse true? where $F$ is cdf and $f$ is pdf I recently ...
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Finding the log Jacobian of a transformation from lower to high dimensions, specifically in the case of normal cdf to Dirichlet?

I have random variables $\mu$ and $\sigma$ that I have transformed and I am interested in finding their joint distribution given the following information I have. Particularly, I need help finding the ...
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1answer
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How do I find the MLE of the APE distribution in R?

The random variable $Y$ is said to have a two-parameter APE distribution, denoted by $\text{APE}(\alpha, \lambda)$, with the shape parameter $\alpha>0$ and scale parameter $\lambda>0$ if the ...
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Efficient Calculation of Moments from arbitrary continuous probability distribution

I want to numerically calculate moments/centralized moments from a continuous probability distribution. The continuous probability distribution is arbitrary which is characterized by a function f(p). ...
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Integration of Logistic distribution in R [migrated]

The probability density function of Logistic distribution $f(x) = e^{-x} (1+e^{-x})^{-2}$ ...
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1answer
27 views

Interpretation of a mixture model

I want to have a mixture model like $\lambda \cdot P(s'|s, a) + (1- \lambda) \cdot P'(s'|s)$ where $P$ and $P'$ are conditional distributions and $\lambda \in [0,1]$ is a weight. I have two questions ...
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I know the 95% confidence interval for ln(x), do I also know the 95% confidence interval of x?

Suppose the 95% confidence interval for $\ln(x)$ is $[l,u]$. Is it true that the 95% CI for $x$ is simply $[e^l, e^u]$? I have the intuition the answer is yes, because $\ln$ is a continuous function. ...
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Proof that $f(t) P(x; \sigma = 1)$ is equivilant to $P(x; \sigma = f(t))$ where $P(x; \sigma)$ is a PDF with a single mode

Consider some function or lineshape, $f(t)$, whose form is known. Also consider some one-parameter PDF, $P(x;\sigma)$ where $\sigma$ is a shape parameter -- and importantly defines the mode of $P(x;\...
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1answer
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Determining whether new data is “in distribution” with training data

I'm hoping to use machine learning to predict chemical properties of various molecules. Many chemistry machine learning research papers that I come across talk about model generalizability issues ...
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How confident can I be that this distribution is random (or not random)?

Here's my background and hypothesis: I have a list of 1000 items, and I choose 100 of them (without replacement). Then I replace the 100 chosen items and repeat this test many times. If the items ...
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Connection between t and F distributions [duplicate]

We have: $Z\sim N_{0,1}$ (standard normal distribution) , $U\sim X^2_{k}$ ($X^2$ distribution, with k df) with $Z \bot U$ Then $X = \frac{Z}{ \sqrt{U /k} } \sim t (k)$ (t-distribution, with k df) I ...
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When calculating weights using inverse probability weighting, should the mean of the distribution of weights =1?

I am trying to calculate stabilized weights using inverse probability weighting by "dropout" from my cohort study to try to account for selection bias due to follow-up. I read from one ...
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1answer
23 views

Optimal combination of biased samplers

Suppose we are interested in the mean $\mu$ of a random variable $X$ but the only way to sample it is from known biased distributions $p_{\lambda}(x)$, such that $\left<{X}\right>_{\lambda} =\...
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Beta Binomial Distribution Derivation- Bayesian

This is from Hoff's Book:First Course in Bayesian Statistics How is p(y|θ) in equation (1) is similar or different than that of equation (2)?. In equation (1) p(y|θ) is treated as joint ...
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1answer
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calculating CDF of kth order statistic

I have recently started probability and statistics on my own. Pls help in understanding below. $x_{(k)}$ is kth smallest random variable from sample of n iids ($x_1 \to x_n$) For calculating CDF of ...
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Are populations related to random variables when discussing parameters?

The term "parameter" (as opposed to a statistic) is defined as a value used to describe a population, but it's also defined as a value used to describe the distribution of a random variable. ...
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What is the conventional way of estimating a population parameter?

I was wondering about the correct way to approach the following question (I am quoting the question as it was originally written): Consider a particle residing at x = 1, y = 1 and the particle begins ...
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Choosing a model for zero-inflated self-reported spending outcome

My outcome variable is individual's self-reported spending ($), which is distributed as follows (rounded to integers). I want to add several independent variables to examine their relationships, and I ...
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R - MLE of modified Champernowne density , using the “nlm” function

I've come across an article (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=704903), in which author wrote about maximum likelihood estimates of parameters in the so called modified Champernowne ...
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1answer
87 views

Change of variables in pdf

I have the joint pdf$$f(x_1,x_2)=x_1e^{-x_1(1+x_2)}I_{(0,\infty)}(x_1)I_{(0,\infty)}(x_2)$$and have to derive the joint pdf of $$Y_1=e^{-X_1}\qquad\text{ and }\quad Y_2=e^{-X_1X_2}$$ I set $x_1=-\ln(...
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What does “A realization of independent copies”? [duplicate]

I read in statistic, "a realization of independent copies" from "Elements of copula modelling with R", and do not understand the meaning. I search and found that it means, the two ...
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1answer
58 views

What does it mean if the distribution of the scores given by a classifier change over time?

I´m currently learning about how to determine if your model keeps performing well or has degraded (particurlarly, for classification problems). My question is, what kind of info can I derivate if I ...
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1answer
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Relationship between the number of moments and Tail of the distribution?

While studying about kurtosis and extreme value theory, I came across the concept of tails of the distribution. So I wanted to ask that why is it such that distribution with higher number of moments ...
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15 views

Approximate or exact distribution of the sum of inverse gamma variables

The random variable ${\left| {H\left( {n,m} \right)} \right|^{ - 2}} \sim Inv - Gamma\left( {{\omega },\frac{\Omega }{{\omega }}} \right)$and independent of each other. What distribution does its sum ...
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R binomial model where DV is a proportion and distribution appears Bimodal

I've been attempting to fit a binomial model to a data set of 1,000,000 accounts where the DV (rr) is a percentage of account balance that been has paid (EX account with total owed of 100 dollars has ...
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Which probability distribution applies to data from like-dislike system (statistics idiot with no intuition)?

Alright, total statistics noob who has to do a bit of comparing samples and statistical inference. I have to compare post interactivity under two conditions, with the null hypothesis that the ...
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2answers
49 views

Expected number of rolls until a number appears $k$ times

Roll a fair die, what is the expected number of rolls until a number appear $k$ times? Not necessarily consecutive. Let $N$ be the number of rolls until a number appear $k$ times. For $k=2$, we know ...
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29 views

Separating signal from noise: noise models in standard deep learning

In a regression setting, one wants to identify some model of a process of interest, based on noisy measurements. The model usually goes like this: $$ y_i = f(x_i, \theta_1) + \varepsilon_i.$$ Here, $...
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1answer
27 views

Computing mean and sd from a frequency histogram (in R)

On page 4 of this document, there is a frequency distribution (histogram) of some students' math scores. You can access the exact same info. shown on page 4 of this document below (in ...