All Questions
Tagged with density or density-function
1,718 questions
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Defining a general distribution over a planar curve
I'm trying to understand how to define an arbritrary distribution over a curve in $\mathbb{R}^2$. To achieve this goal I define a distribution over the unitary interval $[0,1]$ in terms of a ...
- 473
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33
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Is it possible to say which term is grater? [closed]
Compare below:
First summation: 𝑃 ( 𝑋 ≥ 𝑘 − 1 ) for X∼Binomial(n−1,p).
Second summation: P(X≥k) for X∼Binomial(n,p).
.
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3
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2
answers
52
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Monotonic Transformation Preserving Probabilities Intuition
I'm struggling to develop a deep intuition for why monotonic transformations preserve relative probabilities in continuous cases. While I understand the idea at a surface level, particularly for ...
- 191
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13
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Convolution with a pathological distribution part 2
This post is a follow-up to this previous one, based on what I learned from this second one.
Problem Definition
Consider a polygon with vertices $V_1,\dots,V_n \in \mathbb{R}^2$ and let
\begin{aligned}...
- 473
9
votes
1
answer
365
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Defining a uniform distribution over the perimeter of a polygon
Problem definition
Consider a polygon with vertices $V_1,\dots,V_n \in \mathbb{R}^2$ and let
\begin{equation*}
\begin{aligned}
z&=\underbrace{\sum_{j=1}^n \left[(V_{j+1}-V_j) \frac{L}{L_j}\left(\...
- 473
2
votes
1
answer
68
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chi square and exponential distributions
The plot of a single chi square RV approaches infinity at X^2 = 0. Why is that when chi square RVs are added, the resulting pdf plot seems to be at a finite value at Y= 0, where Y = the sum of these ...
- 31
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59
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Copula density estimation and plotting using orthogonal Legendre polynomials
I have been unsuccessfully trying to replicate a copula density plot based on the following steps:
Use a uniform measure on I=[0,1]
Use an orthonormal basis of shifted Legendre polynomials with the ...
user1531131
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26
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Estimating PDF from historical data
Imagine we have a set of points through which we want to fit a probability distribution.
The classic approach would be either:
Parametric estimation (we assume a specific PDF and try to estimate the ...
- 147
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1
answer
45
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How to solve this question about the beta distribution in a Bayesian analysis? [closed]
This question appeared in Prof. Babak Shahbaba's book (Biostatistics With R: An Introduction to Statistics Through Biological Data) in the questions of its chapter 13.
Q4. Suppose that we are ...
- 1,147
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54
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Deriving the pdf of noisy signal: sum of pdf f(s)=1/2+s/2 and a uniformly distributed noise [duplicate]
I am having a hard time to find the pdf of $\widetilde{s}=s+x$.
$s$ has a pdf $f(s)=1/2+s/2$ where $s\in(-1,1)$. $x$ is uniformly distributed over $[-\epsilon, \epsilon]$. I am trying to use the ...
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Simple but important question: how do you write down the formula for the probability density of data in general? [duplicate]
In machine learning many data can be thought of as generated from a probability density function (also called probability distribution).
But most probability textbook only discuss probability density ...
- 671
1
vote
1
answer
62
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Expected Value for Complex-Valued Random Variable
This question is part of Exercise 3.14 in the book The Analysis of Time Series: An Introduction with R, 7th Ed., by Chatfield and Xing.
Problem Statement: Suppose $\theta$ is uniformly distributed ...
- 6,039
2
votes
1
answer
34
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What are some methods to estimate analytical PDF of random variable from an intricate expression of random variables?
Suppose, I have a random variable $y$ distributed as t- distribution
($\mu$) and a random variable $x$ distributed as gamma distribution ($\alpha,\beta$), and a variable $\theta$ distributed as ...
- 189
1
vote
1
answer
83
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Proving that mgf determines distribution via Laplace transform
I am reading this question and the answer provided there about the moment generating function (mgf) and how its uniqueness can be proved via the uniqueness of Laplace transforms. In my book, Measure ...
- 259
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3
answers
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How to show that many functions (a hundred, a thousand) have the same shape an distribution of values over an interval?
I have functions that on iterval [0,1] all seem to look like this:
i.e. they have a zero around 0.4 +ve derivative from zero to 0.4 and around zero or slightly negative derivative up to 1.
I plan to ...
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1
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108
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Does CDF must have value 0 at lowest possible input?
Suppose $F$ is the CDF of a real valued random variable. I know that $F(- \infty) = 0$, because the RV cannot take a value less than that.
But I was thinking of an RV whose value for sure comes from, ...
3
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2
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83
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Exercise on finding probability density function
Let $Y_1$ and $Y_2$ by independent and uniformly distributed over the interval (0, 1). Find the probability density for $U = Y_1/Y_2$:
Solution:
$F_U(u) = P(U \le u) = P(Y_1/Y_2 \le u)$. Looking at ...
- 131
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answers
50
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Distribution of a product of random variables
I have two independent distributions $X$ and $Y$. $X$ is defined by the piecewise CDF
$$F_X(x) = \begin{cases}
F_X^1(x) & x \in (-\infty, a_1)\\
F_X^2(x) & x \in [a_1, a_2)\\
F_X^3(x) & x \...
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votes
1
answer
254
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Pushforward measure for Radon Nikodym equation
Consider the probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and another probability measure $\mu$, on that same space, given by
$$\mu(A)=\int_A f(\omega) \mathbb{P}(d\omega)$$
Now let $X:\...
- 863
3
votes
2
answers
165
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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$. ...
- 81
4
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1
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179
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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}\...
- 473
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41
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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&=...
- 473
8
votes
2
answers
166
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$?
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22
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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 ...
- 13
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1
answer
35
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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) ...
- 311
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43
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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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answers
37
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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 ...
- 1,413
2
votes
1
answer
59
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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 ...
- 451
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23
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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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27
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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.
4
votes
1
answer
543
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 ...
- 304
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25
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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_{-\...
- 303
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2
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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 ...
2
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0
answers
38
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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 $\...
- 525
3
votes
2
answers
127
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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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1
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31
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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+\...
- 161
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49
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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-...
- 273
3
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1
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94
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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 ...
- 225
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51
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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 \...
- 61
1
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1
answer
71
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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
${...
- 171
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1
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84
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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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3
votes
1
answer
155
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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 ...
- 133
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3
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290
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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}
$$
where $a\ge0$ is a real parameter, and $m$ a positive integer?
Since this is ...
- 4,552
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15
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Conditional variance formula for gaussian process classification
I am trying to understand the maths behind scikit learn's Gaussian process classifier. There is a link to the book from which the algorithm was taken. It is a bit involed and there is a particular ...
- 111
1
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25
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Developing a Confidence Interval of Density Functions for Uniform Periods in Seasonal Time Series Data
Suppose I have a set of observational data as a time series where the observations are collected at uniform interval over the course of several years. The data exhibits seasonality over the course of ...
- 11
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25
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Is the following conditional density function equivalent to its unconditional counterpart? [duplicate]
Suppose we have a stochastic series $\{X_t\in\mathbb{R}, t=1,\cdots, T\}$. Further suppose that $G(X_t)=\mathbf{1}_{X_t\geq 0}$ where $\mathbf{1}$ is an indicator function. Can it be concluded that ...
- 1,226
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48
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Converting an integral into a probability of some event
Suppose that $X_1, X_2, .....X_n$ are iid random variables from some continuous distribution $F$. Show that $$\int_0^{\infty}(1-F(s+t))f(s)ds=\mathbb{P}(X_1>X_2+t, X_2>0)$$
$$$$Consider the ...
- 217
1
vote
0
answers
54
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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, $...
5
votes
3
answers
152
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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 ...
- 1,680
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0
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31
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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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