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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Power-law fitting with x-y relationship [closed]
I would like to fit a power-law distribution to data taking into account the x-y-relationship. I tried plfit & powerlaw package for python. But the problem with them is that if I shuffle the y-...
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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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accept reject function x 0<x<1 [duplicate]
hi everyone I'm trying to implement the code in r of accept and reject the function but I don't understand what I should do when my distribution function is not in 0 and 1 as in the case of fx(x)=x
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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}...
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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 ...
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Generating a random number with CDF $P(X \leq c) = 1-1/c$ in the interval $(1, + \infty)$. using uniform distribution
I have a uniform number generator in ($0, 1).$ I want to generate a random number with CDF $P(X \leq c) = 1-1/c$ in the interval $(1, + \infty)$. I know I should apply the inverse of my function to ...
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The Tsallis entropy of generalized Gaussian distribution
I would like to discuss the computation of the Tsallis entropy for the generalized Gaussian distribution. From the paper in the link https://www.sciencedirect.com/science/article/pii/S0167947322000822....
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comparing 2 likelihood values
Are likelihood values (density values) comparable across different types of distributions?
For example, if you have a data point that has a likelihood value of .05 under a normal distribution and .025 ...
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What is $p(y|x)$ given $X=Y+Z$, $Z$ is a standard normal, and $Y$ is a random variable
We have $X=Y+Z$ where $Z$ is a standard normal and $Y$ is a random variable with $p(y)$ as its density. $Y$ and $Z$ are independent.
The conditional probability $p(x|y)$ is obvious to be $\mathcal N_x(...
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Sampling subsets of a given PDF with controllable sum and frequency
I am working with a dataset with $n$ samples and $d$ features for each sample.
I would like to be able to sample "nice" subsets of this dataset with specific properties. Assume that this ...
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On the difference of truncated Gaussian and a new definition
Given a r.v. $X \sim N(0, 1)$, what is the density of $Z = X I(\lvert x \rvert < \lambda)$. I am confused with the truncated Gaussian $Y = X$ if $\lvert X \rvert < \lambda$ otherwise $Y = 0$.
My ...
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Can I use multiple quantile regression to estimate the probability a dependant variable is above / below a certain value?
Let's say I have a dataset of characteristics of newly launched products in a retail environment, and the dependant variable Y is total $ sales in the first year of ...
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If $Z=X+Y$, and I know the probability distribution of $Z$ and $Y$, and $X\perp Y$ how to recover the probability distribution of X?
Suppose I know the distribution of $Z$ and $Y$: $Z\sim F_Z$ with density $f_Z$, $Y\sim F_Y$ with density $f_Y$. Suppose I also know that $Z=X+Y$, where $X$ and $Y$ are independent and the ...
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Calculate mean of density function (example using lognormal distribution)
In a lognormal distribution, the mean is equal to $\exp(\mu + \frac{\sigma^2}{2})$.
I tried to separately calculate this using the definition $E[X] = \int{xf(x)dx}$, where I have 200 $x$ values evenly ...
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Likelihood function is a product of PDFs [duplicate]
I am learning about the likelihood function given iid random variables $X_i$ and realizations $x_i$: $\mathcal{L}(\theta | x) = \prod_{i=1}^n \mathbb{P}(X_i = x_i)$. One thing I am confused about is ...
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Exponential power distribution vs Generalized normal distribution [closed]
statisticshowto and many other sources state that the exponential power distribution and the generalized normal distribution are the same.
However, their pdf really look different:
$$
f_{ep} = (e^{1-e^...
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Confidence interval for unsymmetrical Gaussian Mixure Model PDFs?
Let Y be a vector of observations. A Gaussian Mixure Model (GMM) is fit to the dataset. The distribution can appear unsymmetrical, with different thickness of tails in both sides. What is the best way ...
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Two dimensional random variable with uniform marginal probability density functions [duplicate]
I have access to some data for two variables - let's call them x and y.
In particular, I have the distribution of data separately per each variable, something that allows me to estimate the marginal ...
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Is there a closed form for multi-step ARIMA/ARMA density forecasts conditioned on initial values?/alternatives to this?
I am attempting to create a benchmark for probabilistic forecasting of time series to test other models against and figured that a linear ARIMA/ARMA model would be a good starting point.
I thought ...
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What is the resulting distribution if I merge two different distributions?
The title is not the best but I really do not know how to describe the scenario in a better way.
The context
Consider taking measurements of two different quantities:
The time needed for a car to ...
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PDF of the sine of a wrapped Normal distribution
I have a random variable which is an angle $\Theta$ that follows a wrapped Normal distribution. The angle $\Theta$ has a relatively small variance, so despite having a range from $(-\pi,\pi)$, ...
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Uniform Sampling From the Region Bounded by $\sqrt{x}$, $x=3$, and $y=0$
I want to sample uniformly from the area bounded by $y=0$, $x=3$, and $y=\sqrt{x}$:
If I draw $x$ from $U[0, 3]$ and $y$ from $U[0, \sqrt{x}]$, the density will be higher in the bottom left corner:
...
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how to use the formula in picture for matlab? [closed]
[![the picture is pdf and realibility formula]
1how to use the formula in picture for matlab?
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Calculating PDF conditioned on event
I'm confused about problems where we calculate a PDF conditioned on an event.
Consider this simple problem:
We have two random variables, X and Y, X is uniformly distributed on [a,b], and Y is uniform ...
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Making use of relative density of genomic features between gene sets
I am looking at a particular genomic feature in two sets of genes: set A is a positive control set, where I know this feature is overall enriched in the genomic DNA of these genes, and set B is a (...
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Change of variable formula and probability distributions [duplicate]
I think I have a misconception that I hope someone can help me clarify, I do not have a background in measure theory so that might be the problem.
Say we have two random variables $X$ and $Y$ ...
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Squared norm of linear system proportional to Multivariate Gaussian log-density?
I am reading https://epubs.siam.org/doi/10.1137/140964023, and I got confused by this part:
In the above, it is assumed that $m \geq n$.
If $m = n$, I can see how the above works.
$|| J \theta - y||^...
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Completing the square and marginalizing a multivariate Gaussian [closed]
Edit: This question has been closed for being unrelated although I see similar questions posted here with the same objective, yet not with enough detailed answers or not exactly what I am looking for (...
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Approximate distribution of random variable similar to studentized mean R.V?
It is well known that the distribution of the studentized mean, i.e., $T_0 = \frac{n^{1/2} (\bar{x}- \mu)}{\left(n^{-1} \sum \limits_{i=1}^n (x_i^2 - \bar{x}^2)\right)^{1 / 2}} $, can be approximated (...
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Rate of convergence of covariance of functional
Consider $X_n$ and $Y_n$ to random variable that are bounded in probability.
I know that
$$cov(X_n, Y_n) = O(n^{-1})$$
and that
$$(X_n, Y_n) \rightarrow_d (E_1, E_2)$$
where $E_1$ and $E_2$ are two ...
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First derivative of multivariate normal density with exchangeable correlation structure
As part of a proof, I need to take the first derivative of the log of the following multivariate normal density: $(2\pi)^{-k/2} |\Sigma|^{-1/2} \exp\left(\frac{-1}{2} x'\Sigma^{-1}x\right)$.
In this ...
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Which metric to compare two probability density?
I need to compare two distribution $p$ and $q$. But I don't have access to the distribution $p$, I want to approximate it by distribution $q$ that I construct iteratively by choosing design point. ...
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How to interpret $f_{Y|X}(y|x)$ in the integral of conditional expectation? [duplicate]
$f_X(x)$ gives value of the probability density function of random variable $X$ at point $x$. I am not sure how to wrap my head around $f_{Y|X}(y|x)$; is $Y|X$ still a random variable (sorry for the ...
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How to interpret peaks in probability density function?
If a probability density function (created using kernel density estimation) exhibits peaks (not necessarily the mode), can we infer the presence of clusters or subgroups in the data?
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Finding the probability density function
A random variable $Z$ is obtained as follows. Let $X$ follow $U(0, 1)$, and $Y$ given $X = x$ be Bernoulli with probability of success $x$. If $Y = 1, Z$ is defined to be $X$. Otherwise, the ...
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