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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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Pushforward measure for Radon Nikodym equation

Consider the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and another probability measure $\mu$, on that same space, given by $$\mu(A)=\int_A f(\omega) \mathbb{P}(d\omega)$$ Now let $X:\...
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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}\...
matteogost's user avatar
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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&=...
matteogost's user avatar
8 votes
2 answers
154 views

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 $x$?
Fangzhi Luo's user avatar
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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 ...
Squarepeg's user avatar
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Conditioning once or twice?

Let's say we have two random variables $Z \in \mathcal{Z}$ and $X \in \mathcal{X}$ with joint density $p_{Z,X}(z,x)$ with respect to a base measure. The density is assumed to factor as $$ p_{Z,X}(z,x) ...
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Changing bounds in marginal density

I have the function p(x,y) = 24x for 0<x, x+y<1, x<y. I want to find the marginal density of Y, which means I have to integrate over x. My TA told me I have to split the area I want to ...
Markus J's user avatar
2 votes
0 answers
29 views

To what extent can likelihood methods be used for functional responses?

Let's suppose that we are working with a functional data set, $Y_i(t)$, $Y_i\in L^2[0,1]$, $1\le i\le n$. If we were working with univariate or even multivariate data set, likelihood methods would ...
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Backtransforming a probabilistic forecast?

Let's say that we have a probabilistic forecast for the future percentage return of an asset in the form of a probability density, $\hat{R}_{t+1}$. If our initial goal was to create a probabilistic ...
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Finding the set for random variable transformations

I'm reading through the book "All of Statistics", and in section 2.12, regarding Transformations of Several Random Variables, the author lists three steps for finding the transformation. I ...
David Morton's user avatar
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What is the pdf of the integral of a gaussian process and of the ratio of two gaussian variables?

I need to evaluate the moment functions of a zero mean gaussian process that constitutes the mathematical model of the seismic ground acceleration during an earthquake.
Adrian Daniliuc's user avatar
4 votes
1 answer
527 views

An impossible distribution

Some days ago another user posted a question which was something like this: $$ A \sim U(0,4)$$ $$B \sim U(0,6)$$ $$A - B \sim U(-4,4)$$ The question was originally to find the distribution of A ...
Oscar Flores's user avatar
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Question on the proof step in the theorem 1 of the Gap statistic paper

From the Gap statistic paper, during the proof for the theorem 1, we can see the below equality (p. 422), $\begin{aligned} \operatorname{var}(X) & =\frac{1}{2} \int_{-\infty}^{\infty} \int_{-\...
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Why does re-scaling my density plot using counts change the y-axis so much?

When I make a histogram I get the actual distribution of my samples, with the appropriate counts, but when I try making a density plot the scales go up to 800, and when I try using ...
maglorismyspiritanimal's user avatar
2 votes
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Distribution supported on $(0,\infty)$ for which moments of its truncated distribution are elementary functions of the truncation point and power

I am looking for a distribution with a differentiable PDF $f:(0,\infty)\rightarrow (0,\infty)$ for which for any $\delta>1,z>0$, the two following integrals are finite elementary functions of $\...
cfp's user avatar
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2 answers
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comparing pdf in log scale

In my plot below I am going to compare the pdf of my sample in log scale to the normpdf in log scale . From the plot I can see that the sample pdf roughly follows a standardized normal distribution. ...
V013's user avatar
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1 answer
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Understanding the multivariate normal density proportional

I don't understand the second line of the following equation I get: $$f(x) \propto exp(-\frac{1}{2}(x-\mu)^T \Sigma^{-1}(x-\mu))$$ $$=exp(-\frac{1}{2}x^T\Sigma^{-1}x+\frac{1}{2}x^T\Sigma^{-1}\mu+\...
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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-...
Tim's user avatar
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3 votes
1 answer
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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 ...
SSD's user avatar
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Comparing truncated distributions based on mean and cdf

Let $\tilde{x}$ and $\tilde{y}$ be random variables with pdfs $f_x(x)$ and $f_y(y)$ and cdfs $F_x(x)$ and $F_y(y)$. Given that $E[\tilde{x}] \geq E[\tilde{y}]$ $F_y(c) \geq F_x(c)$ for all $c \in \...
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1 vote
1 answer
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A problem on bivariate random variables

Suppose we have absolutely continuous random vectors $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$. And we have $Y_i=a_iX_i+b_i$, and $a_i>0, b_i\geq 0$ $i=1,2$ . Let ${F}$ be a distribution function such that ${...
Unknown's user avatar
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3 votes
1 answer
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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
Nushko's user avatar
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3 votes
1 answer
110 views

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 ...
Abhinav Tahlani's user avatar
8 votes
3 answers
279 views

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 ...
a06e's user avatar
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0 answers
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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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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 ...
mtp's user avatar
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0 answers
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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 ...
Carl's user avatar
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0 answers
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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 ...
user671269's user avatar
1 vote
0 answers
34 views

A test do check the circularity of a complex variable [closed]

I need a way to measure the circularity of a complex random variable. A complex random variable is circular when its PDF depends only on its magnitude and does not depends on its angle. For example, $...
Ivo Tebexreni's user avatar
6 votes
3 answers
124 views

The height of the pdf is just a relative frequency: is that correct? [duplicate]

I understand that the pdf function is not a probability, and the area under the curve must sum to one. I understand that the height of the pdf function is meaningless, and it is not a probability but ...
Maryam's user avatar
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0 votes
0 answers
30 views

What is density function produced by taking the mean of a finite number of values from a non-normal distribution?

I have a distribution described by the density function 2x*exp(-x^2). I would like to get the distribution that would by produced by taking the average of n observations drawn from this original ...
drmas's user avatar
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2 votes
0 answers
85 views

Find PDF from approximated MGF

I have an array of values of MGF (it is evaluated at some points). The plot of it is shown (blue curve): . Is it possible to find PDF knowing MGF in such form? I tried to fit MGF with some curve (you ...
eMathHelp's user avatar
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1 vote
0 answers
57 views

Joint density of two functions of a uniformly distributed random variable

I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$. I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first ...
johnsmith's user avatar
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0 answers
56 views

What is the dagum type 2 probability distribution function?

I have been searching for dagum type 2 probability distribution function for several hours but all I have found is the cumulative density function of the mentioned distribution which is as follows: $$...
Sepideh Abadpour's user avatar
2 votes
1 answer
164 views

Questions about the conditional Radon-Nikodym derivative

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ and $Y:(\Omega, \mathcal{A})\rightarrow (\mathcal{Y}, \mathcal{G})$ ...
guest1's user avatar
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0 votes
1 answer
152 views

do marginal density functions derived from a joint pdf always integrate to 1 (are they valid pdf's)?

If I have a joint pdf of multiple random variables, say 3 for simplicity, $f_{X,Y,Z}(x,y,z)$, is it true that the marginal density functions derived from that joint probability distribution ( $f_{X}(x)...
Statisticool's user avatar
0 votes
0 answers
59 views

Density of sum of two random variables

Let$(X,Y)$ be an RV of the continous type with PDF $f(x,y)$.Let $Z=X+Y$,then the Convolution of probability distributions told us the PDF of $Z$ is $f_{Z}(z)=\int_{-\infty}^{\infty}f(x,z-x)dx$. If we ...
user553010's user avatar
6 votes
2 answers
132 views

Is there sampling process that admits computing a similarity of two densities when one is intractable?

I have two densities, $p, q$ with sample space $\mathbb{R}^n$, and we can assume both $p,q>0$ (full support). I can compute and sample from $q$. I can compute $p$ up to a constant and I cannot ...
travelingbones's user avatar
2 votes
1 answer
48 views

Interpreting y axis in density plot

200 people were tested, 20 of those were infected. I want to get a posterior distribution for the uncertainty associated with the probability that a person is infected. I do this like this: ...
cvbzxc's user avatar
  • 21
3 votes
1 answer
162 views

Calculating the cumulative distribution function and the probability density function of an interval with ratio of a shorter and longer segment

The interval $[0, 2]$ is divided into two parts by randomly marking a point in $[0, 1]$ according to the rectangular distribution. Let $X$ be the length ratio $L_1/L_2$ of the shorter segment $L_1$ to ...
Ste0l's user avatar
  • 45
2 votes
1 answer
120 views

pdf vs probability vs likelihood [duplicate]

How to compute the log likelihood? Let's take a simple example using a normal distribution and scipy to do the work. Assuming X is the data, and the normal distribution as the model (...
brice rebsamen's user avatar
2 votes
0 answers
35 views

Can a linear combination of two Normal densities be a Normal density? [duplicate]

Consider $f_i(x) = \frac{1}{\sqrt{2\pi}\sigma_{i}}e^{-\frac{1}{2}\left(\frac{x-\mu_{i}}{\sigma_{i}}\right)^{2}},$ $i=1,2$. Define another density by $$f(x) \equiv wf_1(x)+(1-w)f_2(x).$$ Is $f(x)$ also ...
Giorgi Chavchanidze's user avatar
5 votes
1 answer
92 views

Does the PDF $\exp\left(-\frac{x^2}{2}\right) \cosh(\gamma x)$ have a name?

Is there a name for the following probability density function: $$ P(x) \propto \exp\left(-\frac{x^2}{2}\right) \cosh(\gamma x) $$ where $\gamma \ge 0$ and $x\in\mathbb{R}$? Eventually my goal is to ...
a06e's user avatar
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0 votes
1 answer
53 views

Beginner Probability Question: Find PDF and E(X) [closed]

From [1;2] continuous interval we choose 3 numbers randomly. Let $X$ be the minimum between those numbers. Find PDF and Expected value. I fail to understand the problem, since I believe that ...
Nika Kvashali's user avatar
1 vote
1 answer
23 views

Bounded Distribution with specific limits regarding Variance

Im currently looking for a probabilty density function that posesses the following properties Should have range (0,1) $$ \lim_{\sigma \rightarrow 0} f(x) = \delta(1) $$ $$ \lim_{\sigma \rightarrow \...
elson1608's user avatar
1 vote
2 answers
55 views

Help with algebraic steps that my statistics text employed in confirming a conditional distribution [duplicate]

In the page from a statistics book pasted below the authors make the algebraic leap from the LHS to the RHS of the equals sign here: $\large \frac{(x_1 - \mu_1)^2}{\sigma_{11}} - 2\rho_{12} \frac{(x_1 ...
stevedepp's user avatar
2 votes
2 answers
107 views

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: $$f_{X + Y}(x) = \int_{-\infty}^{\infty} f_X(\eta) f_Y(x - \eta) \; d\eta$$ ...
Abram Konzel's user avatar
0 votes
0 answers
767 views

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 ...
KansaiRobot's user avatar
3 votes
0 answers
95 views

Conditional Distribution of Multivariate Gaussian given Linear Inequalities

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 ...
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