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Questions tagged [marginal-distribution]

Marginal probability distributions arise from joint probability measures on product spaces. The marginal distributions are the push-forward measures induced by the coordinate projections.

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Conditional probability for 3 random variables failing

I'm sure I'm doing a wrong assumption here but I can't see it and was hoping someone would point it out. Suppose three continuous random variables $x_t, x_{t-1}$ and $x_0$. We can write the ...
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The marginal distribution and marginal density of e^x

I have the probability density $$f(x,y)= \begin{cases} e^{-x}& \text{for $0\leq y\leq x$}\\ 0&\text{otherwise}\\ \end{cases}$$ Am i correct to assume that the marginal densities are $f_x(x)=xe^...
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Pinning down the joint distribution from marginal distributions?

We know that knowing the marginal distributions of random variables $X$ and $Y$ is not enough to pin down the joint distribution of $\left(X,Y\right)$. But suppose there are three random variables, $X$...
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Distribution of one vector, subject to sample mean and sample covariance constraint.

Let $\{ X_1, \cdots, X_n \} \subset \mathbf{R}^d$ be sampled uniformly from the space of collections of $n$ points in $\mathbf{R}^d$ which satisfy i) $\sum_{i = 1}^n X_i = 0$ ii) $\sum_{i = 1}^n X_i ...
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A random point $X$ from $(0,1)$ is selected. After $X$ is selected, $Y$ is selected from $(0,x^2)$. Find the marginal p.d.f. $f_2(y)$

A random point $X$ from $(0,1)$ is selected. After $X$ is selected, $Y$ is selected from $(0,x^2)$. Find the marginal p.d.f. $f_Y(y)$. Well, I assume a constant probability throughout the region $R=\{...
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Showing a bivariate Gaussian vector has normal components by computing the marginals

Let $( X_1 , X_2 )$ be a bivariate normal random variable. Assume that the correlation coefficient $|\rho[ X_1 , X_2 ] | \ne 1$. Show that $X_1$ and $X_2$ are Normal random variables by calculating ...
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Finding numbers that satisfy these equations - related to marginal distribution

Let $a_1, \ldots , a_n \in [0,1]$ be some numbers such that $\sum\limits_{i=1}^n a_i=m$ for some natural number $m$. I want to prove that there exist nonnegative numbers $b_X$ where the indices $X$ ...
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Everywhere existence of marginals

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a joint PDF, i.e., $\|f\|_{L^1}=1$, which satisfies $f(x,y)>0$ for all $(x,y)\in \mathbb{R^2}$. What is a necessary and sufficient condition under which the ...
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The Problem of Marginalizing Expression (6) in DDIM Papers

I have a question in the process of marginalizing equation (6) in the DDIM paper (https://arxiv.org/pdf/2010.02502.pdf). When the joint distribution is defined like this (eq(6)) $$ q_{\sigma}(\...
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Any copula to force one variable larger than the other in the joint?

I want to transform two known marginal distributions into a joint distribution by using copula. As I understand commonly used copulas cannot make sure that in the joint distribution one variable is ...
Nicolas Fabre's user avatar
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Prove $X$ and $Y$ with $X \sim \text{Unif}(0, 1)$ and $Y = X$ have no joint probability density function

Prove that the two random variables $X$ and $Y$ with $X \sim \text{Unif}(0, 1)$ and $Y = X$ have no joint probability density function (PDF), while each margin has a PDF. Here is my progress, please ...
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Find the marginal density function using integration by substitution

Consider the function $f(x_1,x_2,x_3) = \frac{1}{\sqrt{2\pi^3}}*\Bigl(e^{\frac{1}{2}(x_1^2+x_2^2+x_3^2)}\Bigr)*\Bigl(1+x_1x_2x_3*e^{\frac{1}{2}(x_1^2+x_2^2+x_3^2)}\Bigr)$. I am trying to find its ...
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How do I determine the bounds of integration when finding these marginal pdfs?

I am studying for actuarial Exam P using Finan's notes. I have come across a joint probability problem I do not understand how to solve and the answer key only adds to my confusion. The problem is: ...
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Computing marginal distribution from two independent normally distributed random variables

Problem Given two independent random variables$Z_1 \sim \mathcal{N}(0,1)$ and $Z_1 \sim \mathcal{N}(0,1)$, what are the marginal distributions for $Y_1$ and $Y_2$ in all three cases? (1) $Y_1 = Z_1 + ...
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Different approaches to finding the marginal distribution on surface of sphere with radius $R$ centered at origin.

I asked this question years ago $f_X(x) \neq \frac{2 \pi \sqrt{R^2 - x^2}}{4\pi R^2}$? $X$ belongs to points uniformly distributed on the surface of a sphere., and I'm trying to use 2 other ...
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Finding Marginal Distribution of a multivariate function

Suppose I have a function $$f(x,y,z) \propto x^2yz(1-2x-y-z)$$ where $x>0,y>0, z>0, 2x+y+z<1$ I need to find the marginal functions of X, Y and Z. In normal, two-variable situations, I ...
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Posterior Distribution and James Stein Estimator

Assume $\mathbf{\mu }\sim N\left( 0,I_{r}\right) ,$ where $I_{r}$ is a $% r\times r$ identity matrix, and $\mathbf{y}|\mathbf{\mu }\sim N\left( \mathbf{A\mu },I_{T}\right) ,$ where $\mathbf{A}$ is a $...
Charles Chou's user avatar
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Can marginals determine all probabilities

suppose we are given $n$ binary random variables, $X_1,\dots,X_n$. A probability distribution $P$ assigns all elementary events a probability; $$ P(X_1=x_1\&\ldots \& X_n=x_n)\in[0,1].$$ A ...
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Determining the marginal distribution for a markov chain

Let ${X_n : n = 0, 1, 2, . . .}$ denote a Markov chain with the states $S = {1, 2, 3}$ and transition matrix P given by $$ \begin{bmatrix} 0 & 0.5 & 0.5 \\ 0.1 & 0 & 0.9 \\ 0.8 & 0....
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English translation of Kellerer (1964)

I am wondering if there is an english translation of Kellerer's 1964 paper on existence of joint distributions with particular marginals. The german version is here: https://link.springer.com/content/...
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optimal transport on marginals

Let $\mathcal{M}_1(\mathbb{R}^{n})$ be the space of probability measures on $\mathbb{R}^{n}$. Given two probability distributions $\mu,\nu$ on $\mathbb{R}^{2n}$, let their marginals be denoted $\mu_i,\...
opio's user avatar
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Marginal density functions of $X$ and $Y$

Let $R$ be a region bounded by $y=0$,$y=1$,$x=1$,$y=\sqrt{2x}$. Choose a point $(X, Y)$ uniformly at random from the bounded region. I know that $$f_{X,Y}(x,y) = \frac{1}{\text{area}(R)} = \frac65, \...
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Marginalisation when one variable is discrete and the other is continuous

If we have two discrete random variables $X$ and $Y$, $$p_X(x) = \sum_y p_{XY}(x,y) = \sum_y p_{X|Y}(x|y)p_Y(y) = \sum_y \mathbb{P}(X = x|Y=y) \mathbb P(Y=y)$$ Similarly, if we have two continuous ...
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Does a joint distribution have a unique decomposition into a marginal and conditional distribution?

I believe this is true. The argument is that if we fix $p_{XY}$, then the marginal $p_X$ is immediately fixed. So for any $q_X\neq p_X$, there exists no conditional probability distribution $Q_{Y|X}$ ...
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Schur complement of the marginalized normal covariance matrix given joint Cholesky decomposition

Consider a multivariate normal distribution with covariance matrix $\Sigma$ of size $n \times n$, which can be written in terms of its lower triangular Cholesky decomposition $L$ as $$\Sigma = L \cdot ...
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Can you have a multi-variable Marginal Distribution?

The SHAP algorithm is a commonly used method in Machine Learning to explain black-box models. I'm working on producing my own version of the SHAP algorithm to help my understanding of the method. But ...
Connor's user avatar
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What does a bar "|" between Expectation subscripts mean?

I've seen this notation in the SHAP paper, which extends Shapley values to Machine Learning models to give a form of local explanation. In the paper, on page 5, the author uses the following notation: ...
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Underlying measure of marginal distribution and joint distribution

Given a probability space $(\mathbb{R}^k, \mathcal{B}^k, \mathbb{P}_1)$ and a Markov kernel $\mathbb{P}_{1,2}:\mathbb{R}^k\times\mathcal{B}^l\rightarrow\mathbb{R}$ on a measure space $(\mathbb{R}^l, \...
guest1's user avatar
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Finding the marginal distributions of a Gaussian mixture model. Is it the same as the Gaussian distribution?

Does the marginals of mixtures of Gaussians follow the properties of Gaussian distribution and the definition of marginalization? What I want to do is to obtain the marginal probability density ...
Naomi's user avatar
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Deriving marginal distribution from a joint distribution

Let the joint distribution of a random vector be $$f(x,y)=\begin{cases}1,\enspace 0<x<1, x<y<x+1&\\0, \enspace {\rm otherwise}\end{cases}$$ The marginal pdf of $X$ is $Uni(0,1)$, but ...
In the blind's user avatar
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1 answer
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Radon-Nikodym derivative with respect to product of marginal measures

Let $\mu$ be a (finite if necessary) measure on the product $\sigma$-algebra $\mathcal A_1 \otimes \mathcal A_2$ of two measurable spaces $(\Sigma_1,\mathcal A_1)$, $(\Sigma_2, \mathcal A_2)$. The ...
Michael's user avatar
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Normal distribution for $Z=X+Y$ where $X,Y$ are both normally distributed

I need to find the probability distribution for $Z = X+Y$ where $X \sim \mathcal{N}(x_0,\sigma_x^2)$ and $Y \sim \mathcal{N}(y_0,\sigma_y^2)$. $X$ and $Y$ are independent. In order to do this, we use ...
Tanamas's user avatar
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Determining independence of random variables from joint pdf

I am working on the problem below and have a question about the note that is provided. I know that independent random variables must satisfy the following; $$ f_{XY}(x,y) = f_{_X}(x)f_{_Y}(y) $$ ...
AdamsK's user avatar
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1 answer
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Quantile function, bivariate joint density.

Consider random variables $X,Y$ with joint density function $$ f(x, y)=\frac{1}{2 \pi}\left(1+x^2+y^2\right)^{-3 / 2} $$ I want to find the quantile function of $|Y|$. I have learned how to find the ...
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When are S and T uncorrelated based on the marginal distributions of X and Y? With S = X - Y and T = X + Y

Given two random variables $X$ and $Y$, with $S = X - Y$ and $T = X + Y$. Under what constraint on the marginal distributions of $X$ and $Y$ are $S$ and $T$ uncorrelated. I know that $S$ and $T$ are ...
Bartel Jaap's user avatar
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how to derive a marginal PMF from this cumbersome joint PMF?

I've been learning probability theory recently and got stuck with this problem: knowing that the joint PMF of a 2 dimensional random variable $(X,Y)$ : \begin{equation}P(X=n,Y=m)=\frac{\lambda^np^m(1-...
Erutaner's user avatar
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2 answers
117 views

Compute marginal density given conditional density

Given a continuous random variable $X$ with conditional pdfs $$f_{X=x|Y=0}\qquad \text{and}\qquad f_{X=x|Y=1}$$ and the probability of a discrete random variable $Y$ as $P(Y=0)=0.1,P(Y=1)=0.9$, how ...
Alex99pd's user avatar
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Marginal distribution from joint distribution

I have the joint distribution of two random variables $x,\theta$ $$h(x,\theta)=\frac{1}{2\pi\sigma}\exp\left\{-\frac{1}{2}\left[(x-\theta)^2+\frac{\theta^2}{\sigma^2}\right]\right\}$$ To find the ...
reyna's user avatar
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Calculating marginal mean, variance, and pmf

Suppose that $Y|X \sim\operatorname{Poisson}(cX)$ where $c>0$ is a constant and $X\sim\operatorname{Exp}(1)$ (a) Find the marginal mean and variance of $Y$. (b) Find the marginal pmf of $Y$. For ...
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Distribution of $J$ if $K \sim \operatorname{Poisson}(\mu)$ and $J\mid K = k \sim \operatorname{Bn}(k,p)$.

My working so far: $$ p_K(k) = \frac{\mu^k e^{-\mu}}{k!} \quad \text{and} \quad p_{J|K}(j|k) = {k \choose j}p^j(1-p)^{k-j} $$ Then $$ \begin{aligned} p_{J,K}(j,k) &= p_{J|K}(j|K)p_K(k) \\ &= ...
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Will the maximum entropy joint distribution given a known set of marginal distributions have the maximum plausible support?

Define $[n] = \{1, 2, \cdots, n\}$. Given a distribution $P: \{0, 1\}^{[n]} \to [0, 1]$ and a subset $S \subseteq [n]$, we can define the $S$-marginal of $P$, $P_S: \{0, 1\}^S \to [0, 1]$ as $$P_S(x) =...
Samuel Schlesinger's user avatar
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If $\pi\in P(X^2\times X^2)$ has marginals $\delta_x\times\delta_y$ and $\delta_x\times\nu$, then $\pi=\delta_x\times\delta_y\times\delta_x\times\nu$?

Let $X = \mathbb{R}$ and suppose that the probability measure $\pi\in P(X^2\times X^2)$ has the marginals $\delta_x\times\delta_y$ and $\delta_x\times\nu$, where $x, y \in X$ and $\nu$ is a ...
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Is there a marginal probability measure like a marginal pdf?

Forgive my notation! I don't know much about measure theoretic probability theory. From undergrad probability I learned that a marginal density can be obtained from a joint density function using $$f(...
Matthew Ferrell's user avatar
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1 answer
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Need help with the integral of marginal density function

The joint density of random variables X and Y is given by $$f(x,y)= \begin{cases} \frac{2e^{-2x}}{x} , & 0\le x \lt \infty \ , \ 0\le y \le x \\ 0\quad , & \text{otherwise} \end{cases}$$ ...
ayamelon's user avatar
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multivariate pdf that is sphere-like?

While thinking about probability theory and probability distributions, I realized that I could name several multivariate distributions but could not name any multivariate distributions of genus $g=0.$ ...
zeta space's user avatar
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The projection of n-joint distribution onto 2-joint distribution

Let $P$ be a probability distribution on the product of Polish spaces $X_1\times \cdots \times X_N$. In particular, $X_N=X_j$. Suppose the marginal distribution of $P$ on $(X_i)$ are $(\mu_i)$, where $...
Silentmovie's user avatar
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Proving Lemmas in Gaussian Distribution

I am struggling to prove the following lemmas: How would you suggest me to solve it?
adimath's user avatar
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If $X\sim G(a,b_{1})$ and $Y\sim G(a,b_{2})$, then what will be the density function for U=min(X,X+Y)?

Let $X$ and $Y$ two independent random variables for gamma distributions with common shape parameter $a$ and different rate parameter $b_{1}$ and $b_{2}.$ If $U=\min(X,X+Y),$ then what will be the ...
mojammel's user avatar
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Understanding change of probability density function or proability mass function when "Marginal probability distribution" rule

Nobody actually tell me this simple question so I ask here. For below formula, probability distribution marginalizing, does P(X) and P(X, Y) share same form of PDF or PMF? I assume the PDF or PMF will ...
Young Hyun Yoo's user avatar
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242 views

Marginal density equal to zero everywhere.

I've been working on a two part question on bivariate transformations and marginal densities but am having difficulty finding where I have made a mistake as the final answer for the marginal density ...
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