All Questions
Tagged with density or density-function
448 questions with no upvoted or accepted answers
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Would this way of evaluating this probability be correct?
Suppose I have a discrete variable $S_t$ and a continuous variable $X_t$. Further, suppose I wish to evaluate $P(S_t=s_t)$. Would the below derivations be correct?
\begin{align}
P(S_t=s_t)&=\int P(...
- 1,226
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0
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20
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Mapping Parametric Curves with auxiliary variables
The image below displays an approach of using an auxiliary variable to map the parametric curves of a standard normal pdf and cdf.
In Equation (1), z as r.v. is clearly one-dimensional. However, after ...
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0
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22
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Esitmate of minimal of a function changed after transforming the variable
I want to perform MCMC or HMC for solving minimization problem of a function $f(x)$, then define the corresponding density $$g(x) = \exp\left(-f(x)\right)$$
Because the function of the future apply is ...
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530
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Area between two probability density functions as distance measure
I have two distinct probability density functions, and I would like to find a synthetic measure of how different the two distributions are. Intuitively, it would make sense to me to compute the area ...
- 11
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1
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49
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Estimate at which point a linear model hits a certain value (including probabilities)
I have a simple 1D set of datapoints with a trend, I want to estimate at which point $X_t$ (i.e., at which point in the future) the model will hit a certain threshold $Y_t$:
I can fit a trendline to ...
- 131
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119
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What is the probability mass function of Rock, Paper, Scissors?
I was curious about the statistics behind the game of rock, paper, scissors. Let's say n people are playing, where n is greater than or equal to 2. If when all n people reveal their play and only 1 ...
- 793
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46
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Kernel Density: How do the terms 'global' and 'pilot' translate?
I nearly most of the articles on kernel smoothing or concepts that use kernel density estimations, authors speak of 'pilot' and 'global'.
https://link.springer.com/article/10.1023/A:1008925425102
&...
- 141
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60
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Problem on Discrete Random Variable
Please kindly give a pointer to this question. Generating the discrete variables seems unlikely!
Entrance to a country can be denied for a number of reasons. When someone arrives by air, and their ...
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1
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59
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Method for selecting points from a dataset with known distribution A, so the selected points have known distribution B
I have a dataset (10K points) that was sampled from distribution A (the pdf of distribution A is known).
I want to select a subset of the data (1k points), so the selected points have distribution B (...
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111
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What are examples of symmetric copulas $f(x,y)=f(y,x)$ having relative minima for $f(x,x)$?
In a previous posting on this site RepulsiveBehavior I attempted to detail
a quantum-information-theoretic separability/entanglement problem I am pursuing. Detailed issues of sampling sizes for a data ...
- 161
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66
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Conventions for pmf/pdf
The way I usually see the conventions:
for a continuous random variable $X$, the pdf is denoted $f_{X}(x)$ or $f(x)$,where $x$ is an instance of $X$.
And for a discrete random variable $X$, the pmf ...
- 95
1
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1
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67
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Need help interpreting an answer about the Cauchy distribution and Huygens principle
I'm trying to understand the answer here, which provides a physical interpretation for why the Cauchy distributions mean doesn't exist makes the following statement:
If a unit light source is located ...
- 2,630
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132
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Conditional Density Function on Underlying Exponential
Problem Statement: Suppose $Y_1, Y_2,\dots, Y_n$ are a random sample from an exponential distribution with mean $\theta.$ Let $\displaystyle U=\sum_{i=1}^n Y_i.$ Find the conditional density function ...
- 6,039
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74
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A parameter to differentiate multimodal density plots
I am trying to find a parameter that would summarize the shape of a density plot where:
an insight into the symmetry is given (not a priority); and
how regular/irregular the multi modals are
For ...
- 185
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32
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Dealing with Tall, Thin, Skewed Data
I'm working with a dataset that shows particle movement. There are three broad cases that I see in my data, when they are negatively skewed, approximately normal, or positively skewed. My end goal is ...
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54
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joint and conditional density function
Let $f(x,y,z)$ be the joint density function. I found a reference that this joint density can be written as
$f(x,y,z) = f_1(x|y,z)f_2(y|z)f_3(z)$
I'm wondering if there are alternative forms to ...
- 387
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564
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Probability distribution over a variable? What does it mean?
In this document http://legacydirs.umiacs.umd.edu/~xyang35/files/understanding-variational-lower.pdf at the very first page, it says:
Moreover, uppercase P(X) denotes the probability distribution over ...
- 587
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153
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KL divergence from a family of distributions
The KL divergence between two probability measures $q,p$ is $$KL(q(x)||p(x)) = - \int_{\mathcal{X}} q(x) \log\frac{p(x)}{q(x)} dx.$$
What's the KL divergence between a measure and a set of measures? ...
- 399
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36
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What is the pdf of the square of the product of two correlated normal distributions?
Let $x$ and $y$ denote a bivariate normal random vector with zero means, unit variances and correlation coefficient $\rho$. Then, the pdf of $z=xy$ is known to be
\begin{equation}
f(z) = \frac{1}{\pi\...
- 688
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60
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exponential parameter estimtion from the smallest k-th order statistics
Assume $X_1, X_2, X_3,\ldots,X_n$ are i.i.d. samples from Exp($\lambda$). Assume that the integer $k<n$, is it possible to find a an unbiased estimator for $\lambda$ from the k-th smallest ordered ...
1
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1
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30
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If I have N1 samples from PDF p1 and N2 samples from PDF p2, what PDF describes the combined distribution?
If I have (for example) normal distributions $p_1(x)$ and $p_2(x)$, what PDF (if any) will describe 1000 samples from p1 and 500 samples from p2?
i.e. is there a PDF describing the green histogram $...
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41
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Generating random variates knowing the density function
Let's consider a random variable that is following the distribution with the density function as below:
$$f(x) = \begin{cases} \sum_{i=1}^{\infty} 3.5i(0.3)^{i-1}e^{-5ix} & \text{for $x>0$} \\ ...
- 604
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33
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PDF of function of gamma distribution
If $X = {X_1, \dots, X_n}$ is a sample of independent and identically distributed random variables, each following a Gamma distribution $\Gamma(2, λ)$, with unknown scale parameter $\lambda$, then, ...
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33
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Sampling from an density raised to a non-integer power (also, is this a reasonable thing to think about?)
I have a density that can be written as follows,
$$
p(\mathbf{z}) = \frac{1}{H_f}\prod_iq_i(\mathbf{z})
$$
I want to sample from the density $p(\mathbf{z})$ using samples from each of $q_i(\mathbf{z})$...
- 161
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291
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histogram vs. kernel in density estimation
Assume we have a problem of estimation of a density $f(x)$ over an interval $[0, 1]$. Can a regular histogram (i.e. with equal-sized bins) be viewed as some kind of a kernel?
- 668
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39
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Find the Probability density function of a mean random variable
lets say $Y_{1},\ldots,Y_{n}$ are simple random samples with the PDF: $f_{\theta}(y)=\theta y^{\theta - 1} \mathbb{I}(0 \le y \le 1) $
How can I find the PDF of $\bar{Y}$? is it even possible?
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35
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Get Continuous Distribution from Discrete Variable: Problem 6.77 of Wackerly, Mendenhall, Schaeffer, 5th Ed
Problem Statement:
$\newcommand{\szdp}[1]{\!\left(#1\right)}$
Let $v$ denote the volume of a three-dimensional figure. Let $Y$
denote the number of particles observed in volume $v,$ and assume that $Y$...
- 6,039
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80
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Conditional Probability involving condition on two RVs
Suppose X,Y~exp(2) and are independent. Let W=X+Y.
How do I set up integrals to calculate the following:
f(W|X>Y)
E(W|X>Y)
Thanks!
- 21
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422
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How to calculate confidence interval of the CDF of a non-normal distribution?
For a non-normal distribution, how to calculate the confidence interval of the Cumulative Distribution Function (CDF) of such distribution? Are there any approximations to calculate confidence limits ...
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Is skewness visible in the cumulative distribution function (cdf)?
The following two figures are the pdf's of four parametric distributions and their corresponding cdf's. The most left-ward blue line is clearly not skewed, while the most right-ward orange line is ...
- 4,049
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236
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Conjugate Prior Distribution with a Normally distributed marginal distribution?
The Normal-Inverse-Gamma distribution $(X,\mu,\sigma^2)\sim NIG(\mu_0, \nu, \alpha, \beta)$ is the conjugate prior for the Normal distribution. However, this would correspond to the marginal ...
- 421
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47
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How do I evaluate this Integral?
I’m reading Glen Cowan’s book on statistical data analysis and was stuck on an integral that has to do with crv transformations. I’m not able to evaluate the integral
$$
g(a)da = \int_{x(a)}^{x(a)+|\...
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48
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Where does $X_n$ converge to?
Let $ X_1, X_2, X_3, \ldots $ be independent random variables and let $ X_n $ have a probability density fucntion (PDF) defined by
$ f_{X_n}(x) \quad=\quad \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\...
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534
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PDF of Beta Distribution written as exponential family form
I am trying to write the pdf of a beta random variable in its biparametric canonic form such as:
Function 1
$$
f_Y(y; \theta, \phi) = exp \{ \phi[y \theta - b(\theta)] + c(y, \phi) \} \mathbb{1}_A(...
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If $Y$ is continuous and $X$ is discrete, how to write the joint density of $(Y,X)$?
If $Y$ is continuous and $X$ is discrete with a finite number of points in the support, how to write the joint density of $(Y,X)$?
For example, to write the joint density function evaluated at $(Y,X)=(...
- 3,050
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53
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Approximate PDF function from "how many in each range" data
I have the following data which represent how many graduates (out of 578) have an average grade in each range:
$58$ with average grade in the range $[5, 5.99]$
$336$ with average grade in the range $[...
- 111
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253
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How to compare two set of PMFs?
I'm facing some challenge and I don't know the correct approach for this.
I'm having two sets of PMFs $S_1, S_2$ and I need to compare (distance like Jensen–Shannon) $S_1$ with $S_2$.
What's the best ...
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Are the following model assumptions on a data stream too restrictive?
Suppose that you were to model a "generic" continuous-time real-world data signal $X$ taking values in a bounded continuum $K\subset\mathbb{R}^d$ (e.g. the body temperature of a patient or ...
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If I want to model a bivariate distribution that is symmetric about (0,0) using copula, what copulas can I use?
If I want to model bivariate data $\{X_i,Y_i\}_{i=1}^{n}$ using copula. The true joint density of $(X,Y)$ denoted as $f_{XY}(,)$ is unknown, but I know it's symmetric about (0,0) in the sense that
$f_{...
- 3,050
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118
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PDF of squared random variables
i have complex gaussian random variable given as $h\sim\mathcal{C}\mathcal{N}(0,\sigma^{2})$. And i had $Y = |h|^{2}$. So what should be pdf of $Y$. I understood that $Y$ is exponential random ...
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62
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How to define and plot a distribution function in python?
I want to define a distribution function (gaussian or skewed,...), the X axis is from 0 to 255. I have the mode which is located at the point 100 and i have two points (40, 170) that i consider ...
- 43
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52
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Find $x_0$ that satifies $\mathbb{P}(X \leq x_0) = 0.75$
Suppose that $X$ is a continuous random variable with probability density function:
$$\begin{cases}
x & 0 \leq x < 1, \\[6pt]
2-x & 1 \leq x < 2, \\[6pt]
0 & \text{otherwise}. \\...
- 111
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14
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bivariate conditional joint sensor model
I am struggling to find $P\left( V_t | z \right)$ from $P\left( V_t | z , V_p \right)$. Here $z$ and $V_p$ are independent variables.
...
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28
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Can I perturb a sample drawn from a certain gaussian such that it becomes drawn from another gaussian?
Given known $v_1$ and $v_2$ such that $v_2 > v_1$, and an unknown $\mu$:
I have a single sample drawn from the distribution $\mathcal{N}(\mu, v_1)$.
Can I somehow perturb this sample in a sound ...
- 121
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75
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PDF of sum of stationary processes
I would like to obtain formulas for the sum of random processes $U(\omega,t), V(\omega,t)$:
Sum of two signals: $G(\omega,t) = U(\omega,t) + V(\omega,t)$
Case 1: $U,V$ are also jointly stationary
$...
- 111
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47
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Integrate out the binary indicator variable in a two-sample ANOVA
I have two sets of data, A and B, that have unequal sizes, and I want to compare their means. The standard approach would be to do a t-test. Getting a little more sophisticated, we can think of that t-...
- 67.2k
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281
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Can we sample from the wrapped normal distribution and evaluate the density of the sample simultaneously?
In a computer program (written in C++), given $x\in[0,1)$ and $\sigma>0$, I need to sample $y$ from the wrapped normal distribution $\mathcal W_{x,\:\sigma^2}$ with mean $x$ and variance $\sigma^2$ ...
- 213
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37
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Conditional and density probability (normal distribution)
I am trying to solve the following problem:
Suppose that $\mu\sim N(1,4)$ and $Y|\mu\sim N(\mu,1)$. Show that:
$$\begin{bmatrix}Y \\ \mu \end{bmatrix} \sim N\bigg(\begin{bmatrix}1 \\ 1 \end{bmatrix},...
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0
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77
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Calculating a baseline probability model for images
I'm a newbie to statistics, so I apologize if this question is trivial.
I'm trying to build a distribution that can predict a specific set of images. But first, I need a baseline - so, I decided to ...
- 123
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117
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Uniqueness of change of variable function
Let $X$ and $Y$ be continuous random variables with probability density function as $p_x(X)$ and $p_y(Y)$. If $X$ and $Y$ are related by an invertible function $f$ as $f(X)=Y$, then using change of ...
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