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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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 $\...
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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. ...
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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-...
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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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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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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
${...
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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}
$$
where $a\ge0$ is a real parameter, and $m$ a positive integer?
Since this is ...
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Conditional variance formula for gaussian process classification
I am trying to understand the maths behind scikit learn's Gaussian process classifier. There is a link to the book from which the algorithm was taken. It is a bit involed and there is a particular ...
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Developing a Confidence Interval of Density Functions for Uniform Periods in Seasonal Time Series Data
Suppose I have a set of observational data as a time series where the observations are collected at uniform interval over the course of several years. The data exhibits seasonality over the course of ...
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Is the following conditional density function equivalent to its unconditional counterpart? [duplicate]
Suppose we have a stochastic series $\{X_t\in\mathbb{R}, t=1,\cdots, T\}$. Further suppose that $G(X_t)=\mathbf{1}_{X_t\geq 0}$ where $\mathbf{1}$ is an indicator function. Can it be concluded that ...
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Converting an integral into a probability of some event
Suppose that $X_1, X_2, .....X_n$ are iid random variables from some continuous distribution $F$. Show that $$\int_0^{\infty}(1-F(s+t))f(s)ds=\mathbb{P}(X_1>X_2+t, X_2>0)$$
$$$$Consider the ...
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A test do check the circularity of a complex variable [closed]
I need a way to measure the circularity of a complex random variable. A complex random variable is circular when its PDF depends only on its magnitude and does not depends on its angle.
For example, $...
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The height of the pdf is just a relative frequency: is that correct? [duplicate]
I understand that the pdf function is not a probability, and the area under the curve must sum to one. I understand that the height of the pdf function is meaningless, and it is not a probability but ...
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What is density function produced by taking the mean of a finite number of values from a non-normal distribution?
I have a distribution described by the density function 2x*exp(-x^2). I would like to get the distribution that would by produced by taking the average of n observations drawn from this original ...
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Find PDF from approximated MGF
I have an array of values of MGF (it is evaluated at some points).
The plot of it is shown (blue curve): .
Is it possible to find PDF knowing MGF in such form?
I tried to fit MGF with some curve (you ...
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Joint density of two functions of a uniformly distributed random variable
I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$.
I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first ...
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What is the dagum type 2 probability distribution function?
I have been searching for dagum type 2 probability distribution function for several hours but all I have found is the cumulative density function of the mentioned distribution which is as follows:
$$...
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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 $Y:(\Omega, \mathcal{A})\rightarrow (\mathcal{Y}, \mathcal{G})$ ...
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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 ( $f_{X}(x)...
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Density of sum of two random variables
Let$(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 ...
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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 ...
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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:
...
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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 ...
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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 (...
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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 ...
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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 ...
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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 ...
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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)
$$
\lim_{\sigma \rightarrow 0} f(x) = \delta(1)
$$
$$
\lim_{\sigma \rightarrow \...
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Help with algebraic steps that my statistics text employed in confirming a conditional distribution [duplicate]
In the page from a statistics book pasted below the authors make the algebraic leap from the LHS to the RHS of the equals sign here:
$\large \frac{(x_1 - \mu_1)^2}{\sigma_{11}} - 2\rho_{12} \frac{(x_1 ...
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If $X$ is a random variable, why is the PDF of $X + X$ not the same as the PDF of $2X$?
Background:
According to Wikipedia, the PDF of the sum of two random variables $X$ and $Y$ is given by the convolution:
$$f_{X + Y}(x) = \int_{-\infty}^{\infty} f_X(\eta) f_Y(x - \eta) \; d\eta$$
...
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Explanation of what a density plot is [duplicate]
I have been working with histograms so far. I understand what they show.
I am trying to understand what density plots are. In this tutorial it says
The curve (of the density plot)represents the ...
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Conditional Distribution of Multivariate Gaussian given Linear Inequalities
Consider a multivariate Gaussian $Y\sim\mathcal{N}(\mu,\Sigma)$ of dimension $n$. For fixed $c\in\mathbb{R}^n, A\in\mathbb{R}^{m\times n}$ and $c\in\mathbb{R^m}$, what is the conditional distribution ...
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How can I create realistic noisy data from distributions?
I want to create synthetic data from stitched distributions in order to test some models on them (for example Gaussian stitched with a GPD at quantile q).
I'm currently simply sampling N*q points from ...
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Correspondence between the "density function of a probability measure" and the "probability density function" (PDF)
Question. If there is a one to one correspondence between a "borel probability measure" on the line $\mathbb{R}$ and a "cumulative distribution function" (CDF) (please see on page ...
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Any closed-form solution to this integral (multivariate exponential)?
Here is the probability density function (unnormalized) of a covariance matrix: (from a Bayesian perspective):
$$
f(\boldsymbol{V})\propto \det(\boldsymbol{V})^{-\frac{N+J+1}{2}}\int_{\mathbb{R}^{K}}\...
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What is the cross product of two probability distributions $P \times Q$?
In many papers on machine learning and statistics, I encounter the following notation
Let $P$ be a distribution, and let $Q$ be another distribution.
Then the author creates an object $P \times Q$
...
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Is this probability density function computable (seems like a generalized multivariate t-distribution)?
I have obtained an unnormalized probability density function (pdf) for the coefficients of a linear regression model:
$$
f(\boldsymbol{\beta})\propto \mathrm{det}\left [\sum_{i=1}^{N}
(\boldsymbol{y}...
4
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1
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Markov's inequality intuitions
Can someone explain intuitively how Markov's inequality was derived? It seems plausible, but looking a it, I can't 'see' how it's true.
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What is the conditions to use a weights argument to a linear model, when the dependent variable is a proportion?
My data consists of the independent variable (x) which is slope gradient (°) and the dependent variable (y) is collar GPS point density/km². For each slope gradient, the independent variable was ...