Questions tagged [density-function]

Probability density function (PDF) of a continuous random variable gives the relative probability for each of its possible values. Use this tag for discrete probability mass functions (PMFs) too.

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Distribution/expected length of the shortest path in infinite random geometric graphs

Consider an infinite random geometric graph $G(\rho,d)$ in which vertices are uniformly and independently scattered over the 2D plane with density $\rho$ and edges connect the vertices that are closer ...
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What does it mean for a probability distribution to not have a density function?

I understand the distinction between probability mass and density functions. But I don't understand what it means for a continuous random variable to have a probability distribution but not a density. ...
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Spinograms vs. conditional densityplots

I have a binary response variable (hail) and multiple continuous predictor variables. My aim is to understand the linear/non-linear relationship of the predictors to the response to be able to justify ...
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How can a probability densitiy be estimated based on the maximum entropy principle, with constraints in the order statistics?

Let's say we are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$. The ...
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Expectation of a strictly increasing function

Assume that $X_1$ and $X_2$ are two i.i.d. random variables with pdf $f$. Also, assume that $a$ and $b$ are two fixed real numbers such that $a>b$. If $g$ is a strictly increasing function, do I ...
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What's the distribution of $|y-z|^2/|y-\bar{y}|^2$ for vectors with i.i.d. standard normal coordinates?

Let $y_1, y_2, \ldots, y_n$ and $z_1, z_2, \ldots, z_n$ be samples of size $n$ of a normal distribution $\mathcal{N}(0,1)$. My goal is to find the distribution of $$\frac{\sum_{i=1}^n (y_i - z_i)^2}{\...
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Integrating the inverse-Wishart density

It is alleged in this question and in the Wikipedia article and elsewhere that the density function for the inverse-Wishart distribution with $n$ degrees of freedom on $p\times p$ positive-definite ...
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Cumulative distribution function for the product of two random variables

Given two random variables $x, y$, each with the probability distribution functions $p_x(x)$, $p_y(y)$, then if $z = xy$, then $p_z(z) = \int p_x(x)p_y(z/x)\frac{1}{|x|}dx $. Is there a similar proof ...
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What is the density of a markov chain when its transition probabilities have densities with respect to different measures?

I have a homogenous, discrete time Markov process, $(X_n)_{n\geq 0}$, with state space $\mathbb R_+$. Its transition probabilities have a density, $f(x_n\mid x_{n-1})$, with respect to the measure $\...
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How to improve estimation of a deconvolved density

I have the following problem: Y = X + e with Y = Total reaction time (noisy signal) X = selection time (signal) e = discrimination time (noise) I am interestend in the distribution for X and ...
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Joint distribution of two distances

Suppose there are three points in 3D space, each with coordinates $A_i=(X_i,Y_i,Z_i)\leadsto \mathcal{N}(\mu_i,\tau^2\mathbb{I}_3)$. We compute the distance between the three points, e.g. $D_{ij} = \|...
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Finding the probability density function of Hotelling's T-squared distribution

The following image is seen on wikipedia when searching for Hotelling's T-squared distribution This is apparently the pdf of the Hotelling T-squared distribution at different parameters. However, I ...
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Point process - intensity function vs probability density function

Suppose we have a point process in $\mathbb{R}$ with intensity $\lambda(x)$. Then, for a given compact set ${ S}$ we have $$\Lambda({ S})=\int_{\rm S} \lambda(x) \, dx,$$ where $\Lambda({ S})$ is ...
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Automatic fitting of normalization constant as a parameter in noise contrastive estimation

In the paper on Noise Contrastive Estimation, the authors define a parameterized density function $p_m^0\left(x;\alpha\right)$ to estimate the unnormalized PDF of the data, and then further define a ...
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Sum of truncated Gammas and degenerate

I have a variable $X$ which I am modelling with a mixture model: $$\begin{aligned} (X|A) &\sim \mathbb{1}_{0 \leq x < w \cdot m} \cdot \frac{\text{Gamma}(\alpha,0,\beta / m)}{k_1} \\ (X|B) &...
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Testing for Normality (CDF)

I was reading an article about using the CDF to calculate the area between 2 points on the normal curve. They gave a sample of 7 for illustration purposes: ...
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Problem involving P.D.F. containing an indicator variable

Let $X_1, X_2, \ldots$ be independently and identically distributed random variables with probability density functions: $$f(x) = \alpha \;x^{-(\alpha+1)} \; I_{(x>1)}, \; \; \alpha > 0.$$ For ...
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CDF of the ratio of two correlated $\chi^2$ random variables

It is well known that the sum of a series $m$ of squared standard independent normal random variables follows a $\chi^2$ dstribution with $m$ degrees of freedom. It is also true that the ratio of two ...
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Where is the maximum bias and variance in a histogram as non-parametric density estimator?

I am a little bit confused about bias and variance of non-parametric density estimators and hope you can help me. Assuming a constant bandwidth and sample size, I am wondering at which points of the ...
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Geometric construction of copula - question regarding C-volume

I am learning about copula's, using Nelsen's book, and more specifically about the geometric method of constructing copula's. The problem is replicated in the following link: http://www.stat.ubc.ca/...
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Maximum likelihood estimation involving both probabilities and probability densities

Note: based on suggestions in the comments, I have rewritten this question. Please refer to the history for the original version. In general my question regards how to compute likelihoods in mixed ...
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Expectation of density ratio of two iid variables

Let $X \sim N(0,1)$ and $Y \sim N(0,1)$ be independent RVs and let $f$ be their density function. I'd like to compute the expectation of the density ratio \begin{align} \mathbb{E}\left[\frac{f(X)}{f(Y)...
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521 views

Confusion related to Parzen window

I was going through this tutorial related to Parzen window at http://www.cs.utah.edu/~suyash/Dissertation_html/node11.html. However, I have some confusion related to Parzen window with gaussian kernel ...
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Density function of a dependent sum of products of normal random variables

Say we have a random variable $$ X = A_0 A_1 + A_0 A_2 + A_1 A_2, $$ which consists of normally distributed independent random variables $A_0, A_1, A_2 \sim \mathcal{N}(0,1)$ with probability ...
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3 votes
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212 views

What is p(data) in image generation

In the context of image generation architectures such as VAEs or GANs (say we are using mnist digits) what do we mean by probability distribution of the data? Just to clarify this question and make it ...
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Integral of difference of density functions of two Continuous Random Variables goes to 0

The problem says : Let $(X_n)_{n=1}^\infty$ be a sequence of continuous random variables with probability density functions $(f_n)_{n=1}^\infty$ , and let $X$ be another continuous random variable ...
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3 votes
1 answer
155 views

Is limiting density of discrete points (LDDP) equivalent to negative KL-divergence?

Is limiting density of discrete points (LDDP), which is a corrected version of differential entropy, equivalent to the negative KL-divergence (or relative entropy) between a density function $m(x)$ ...
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Is it possible for a distribution to have a non-zero probability of generating a value with zero probability density?

I am reading the book "An Elementary Introduction to Statistical Learning Theory" and there is a sketch of a proof (Section 8.4) for the universal consistency of kernel rules for binary ...
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Fitting Pareto distribution to data example in SciPy

In docs.scipy.org there's code to sample data from a Pareto distribution and then fit a curve on top of the sampled data. I could understand most of the code snippet except the term ...
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What methods are there for estimating distributions based on histograms?

I recently worked on a consulting project where a client wanted to estimate gamma and weibull distributions based purely on histograms rather than raw-data. I have never worked with problems like that ...
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3 votes
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Calculating a Confidence Interval for a Proportion for a Sample of Different Size

I'm interested in a (preferably analytic) solution or approximation to the following problem: Let $s_1$ be a sample from an unknown distribution of size $N_1$ and with proportion of successes $p_1$. ...
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3 votes
1 answer
107 views

Finding expression of $n$-th derivative, when $n$ is large

For completeness, assume $C$ is an Archimedean copula with some generator function $\varphi$, which is usually assumed to have nice properties. It is known that $$ C(u_1, u_2, \ldots, u_n)=\varphi^{-1}...
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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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3 votes
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448 views

Constructing a joint distribution from pairwise bivariate marginal distributions?

It's fairly well-known that given univariate distribution functions $F_X, F_Y, F_Z$, one can construct the joint distribution $F_{(X, Y, Z)}(x, y, z) = C(F_{X}(x), F_{Y}(y), F_{Z}(z))$, where $C$ is ...
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3 votes
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275 views

Measure-theoretic derivation of change of variables formula for probability density functions?

Assume we have a $2$-dimensional sample space $(\Omega, B, P_\Omega)$, with $\Omega =\mathbb R^2$ with borel measure and probability measure $P$, where the axes are simply equal to random variables $...
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3 votes
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Finding PDF from CDF

I just got really unsure, can someone confirm/rectify? I have the CDF defined as $F(x)= \begin{cases}0, &\text{if}~x < 0,\\ 4x^2 &\text{if}~ 0 \leq x < \frac{1}{4} \\ 1-\frac{4}{3}(1-x)^...
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Does the integral of the probability density function squared mean something?

I've been searching but I couldn't find on the internet if there's any significance for the integral of the pdf squared: $\int_\mathbb{R} f^2(x)$ That's because, as an alternative of Shannon entropy,...
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PDF of function $\mbox{max}(x,f(x))$

This is from the book Python for Probability, Statistics, and Machine Learning. It's a good book for people that already have a math background, I believe. Anyway, there is something I don't ...
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Why is the Kolmogorov–Smirnov (KS) test more popular than the Overlapping Coefficient (OVL)?

The Overlapping Coefficient (OVL) measures the common area between two Probability Density Functions (see this question for more details). Intuitively, this seems like a good way to gauge the ...
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Statistics of Intervals

Question Dear all, Assume you ask N persons to set their personal interval of acceptance (e.g. interval ranging from 0 to 100% for the power of a speaker system) which they would rate as enjoyable. ...
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Is the Gaussian distribution the only statistical distribution fully determined by the mean and variance?

I've read that the Gaussian marginal is fully determined by the mean and variance. What does this mean in reality? If we consider a Gaussian marginal PDF is given by $$ \pi_G(\xi|\mu,\sigma) = {1\...
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3 votes
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128 views

Sampling from $f(x)$ given approximation $g(x)$

(After some pondering, what I really wanted to ask is how to incorporate prior information about $f$ into a sampling method - see this question.) Suppose you want to draw samples from an (...
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Probability Density Function of a linear combination of 2 dependent random variables, when joint density is known

Let's say there are two dependent random variables $X$ and $Y$ with joint density function $f$. What is the PDF of the weighted sum of these two variables, $Z = aX + bY$? Thanks in advance for any ...
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3 votes
0 answers
181 views

Bias of an estimated Gaussian density

I have an iid sample, $X_1,\dots,X_N \in R^d$, from a multivariate normal density with mean $\mu$ and covariance matrix $\Sigma$. I am estimating the density $p(y) = N(y| \mu, \Sigma)$, using $\hat{...
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3 votes
0 answers
238 views

Derivation of likelihood function for latent variable model made explicit

I am trying to make the steps deriving the likelihood function for the following latent variable model as explicit as possible: $$Y^0=X\beta + u$$ where $$u \sim NID(0,\sigma^2).$$ The observed data ...
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67 views

Distribution with fixed mean and closest to a given distribution

I was wondering if this problem has been tackled in some way in the probability/functional analysis literature: Given a pdf $f$ such that the expectation is zero and $\mu\in\mathbb R$, find the ...
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Multivariate distribution for products of random variables

Suppose I have an $n$-dimensional complex, zero mean normal distribution with covariance matrix $\Sigma$, which is not diagonal. Denoting each of the random variables as $x_1, \dots ,x_n$ I would ...
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Estimating time-lagged mutual information for two signal samples

This is an attempt to reproduce Moon et al. 1995, and the author's copy can be obtained through here. As a benchmark, we estimate the time-lagged mutual information of a simple sine signal $\sin(0.02\...
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3 votes
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48 views

Error bounds when approximating densities

I was curious whether it is possible to obtain approximation error bounds on continuous densities from approximation error bounds of random variables. To make my question more precise: We consider ...
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84 views

Formalizing pdf using both discrete and continuous densities

I'm trying to formalize the probability density function for a rather simple process, but I'm having difficulty writing it precisely. Specifically, consider simulating a 1-D Gaussian random walk ...
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