# 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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### Estimating Smooth Density Field from Limited Sampled Data

I want to estimate a “density field”, specifically $P(y|x, m)$, for binary labels $y$ associated with 2D points characterized by spatial coordinates $m$ and additional spatio-temporal features $x$. ...
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### Uniform distribution over a triangle

Problem Consider a triangle $T$ with vertices $V_1,V_2,V_3 \in \mathbb{R}^2$ and let \begin{equation*}\begin{aligned} y&=z+v\\ v&\sim\mathcal{N}(0, R)\\ z&\sim\mathcal{U}(T) \end{aligned}\...
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### Uniform density over 2 segments [duplicate]

Background Let $V_1, V_2 \in \mathbb{R}^2$ be the vertices of a segment and let $z$ be uniformly distributed over that segment. Now consider the random vector \begin{equation*} \begin{aligned} y&=...
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### Sum of density functions

Consider four pdf $f_1(x), \ldots, f_4(x)$. For any $x$, $f_1(x) \neq \cdots \neq f_4(x)$. Can we prove that $f_1(x) + f_2(x) \neq f_3(x) + f_4(x)$ for some $x$?
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### Determining Distribution for Conditional Probability

I have that the conditional probability density of $Y|X$ is as such $f_{Y | X} \propto x^{y - 1}(1-x)^{n-y-1}\alpha^{n-y}\beta^{y}$ where $\alpha, \beta$ are constants in $(0, 1)$, $x$ is a random ...
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### Implementing Convolution Function for Gaussian Kernel in Python for PDF Estimation

I am currently working on estimating a probability density function (PDF) nonparametrically using a Gaussian kernel. My goal is to determine the optimal bandwidth $h$ that minimizes the cross-...
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### Name of PDF? - projecting uniform probability distribution on the unit circle to the x-axis

Consider a uniform probability distribution on a circle of radius r, i.e. $\{(x,y) \in \mathbb{R}^2: x^2 + y^2 = r^2 \}$.If we wish to project onto the x-axis, we can consider each point on the circle ...
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### Density Forecasts with GAMLSS

Does someone know the function to create density forecasts within the GAMLSS Package? The predict. Formula is not the right one. Predict do Point Forecasts
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### Zero variance but non-zero skewness

I was thinking of a hypothetical distribution where the mean(first cumulant) is non-zero, second cumulant(variance) is zero, and the third cumulant(skewness) is non-zero. The higher order cumulants ...
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### Sampling from $P(x) \propto \cosh^{m}(a x) e^{-x^{2}/2}$

Is there an efficient algorithm to draw samples $x \sim P(x)$ from the PDF: $$P(x) \propto \cosh^{m}(a x) e^{-x^{2}/2}$$ where $a\ge0$ is a real parameter, and $m$ a positive integer? Since this is ...
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### Conditional variance formula for gaussian process classification

I am trying to understand the maths behind scikit learn's Gaussian process classifier. There is a link to the book from which the algorithm was taken. It is a bit involed and there is a particular ...
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1 vote
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### Developing a Confidence Interval of Density Functions for Uniform Periods in Seasonal Time Series Data

Suppose I have a set of observational data as a time series where the observations are collected at uniform interval over the course of several years. The data exhibits seasonality over the course of ...
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### Is the following conditional density function equivalent to its unconditional counterpart? [duplicate]

Suppose we have a stochastic series $\{X_t\in\mathbb{R}, t=1,\cdots, T\}$. Further suppose that $G(X_t)=\mathbf{1}_{X_t\geq 0}$ where $\mathbf{1}$ is an indicator function. Can it be concluded that ...
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### Converting an integral into a probability of some event

Suppose that $X_1, X_2, .....X_n$ are iid random variables from some continuous distribution $F$. Show that $$\int_0^{\infty}(1-F(s+t))f(s)ds=\mathbb{P}(X_1>X_2+t, X_2>0)$$ Consider the ...
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### If $X$ is a random variable, why is the PDF of $X + X$ not the same as the PDF of $2X$?

Background: According to Wikipedia, the PDF of the sum of two random variables $X$ and $Y$ is given by the convolution: $$f_{X + Y}(x) = \int_{-\infty}^{\infty} f_X(\eta) f_Y(x - \eta) \; d\eta$$ ...
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### Explanation of what a density plot is [duplicate]

I have been working with histograms so far. I understand what they show. I am trying to understand what density plots are. In this tutorial it says The curve (of the density plot)represents the ...
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Consider a multivariate Gaussian $Y\sim\mathcal{N}(\mu,\Sigma)$ of dimension $n$. For fixed $c\in\mathbb{R}^n, A\in\mathbb{R}^{m\times n}$ and $c\in\mathbb{R^m}$, what is the conditional distribution ...