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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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 $...
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Marginal Filtering and Smoothing Distribution in Hidden Markov Model
I want to know the calculation of marginal filtering distribution and smoothing distribution from the joint distribution. For example, define the $x_t$ as the hidden variable in $t$-th step and $y_t$ ...
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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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Does weighted sum of normal densities act like a normal density of the weighted sum?
During a part of my thesis I came across the following problem:
Let $\Theta \sim \nu(\theta)$ be a discrete RV, with PMF $\nu(\theta)$ that represents the prior probability function for our ...
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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
$$
\begin{bmatrix}
0 & 0.5 & 0.5 \\
0.1 & 0 & 0.9 \\
0.8 & 0....
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English translation of Kellerer (1964)
I am wondering if there is an english translation of Kellerer's 1964 paper on existence of joint distributions with particular marginals. The german version is here: https://link.springer.com/content/...
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Can I find the marginal distribution by fitting a distribution to the event dimension, rather than integrating over the variables?
I have a 6-dimensional data which I fitted with a 6-dimensional Gaussian mixture model and I would like to obtain the marginal distribution of all 1-dimensional events. However, I find it difficult to ...
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optimal transport on marginals
Let $\mathcal{M}_1(\mathbb{R}^{n})$ be the space of probability measures on $\mathbb{R}^{n}$.
Given two probability distributions $\mu,\nu$ on $\mathbb{R}^{2n}$, let their marginals be denoted $\mu_i,\...
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Marginal density functions of $X$ and $Y$
Let $R$ be a region bounded by $y=0$,$y=1$,$x=1$,$y=\sqrt{2x}$. Choose a point $(X, Y)$ uniformly at random from the bounded region. I know that $$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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Finding the probabity of a minimum from a joint marginal probabity
Suppose I have the following marginal
$P(Y) = \sum\limits_{x} P(Y|X)P(X)$
Assuming $Y_1 \cdots Y_n$ are iid and same for $X$
Thus, to obtain $Y=i$ we compute :
$P(Y=i) = \sum\limits_{x} P(Y=i|X)P(X)$
...
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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
$$\Sigma = L \cdot ...
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Can you have a multi-variable Marginal Distribution?
The SHAP algorithm is a commonly used method in Machine Learning to explain black-box models. I'm working on producing my own version of the SHAP algorithm to help my understanding of the method. But ...
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What does a bar "|" between Expectation subscripts mean?
I've seen this notation in the SHAP paper, which extends Shapley values to Machine Learning models to give a form of local explanation.
In the paper, on page 5, the author uses the following notation:
...
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Underlying measure of marginal distribution and joint distribution
Given a probability space $(\mathbb{R}^k, \mathcal{B}^k, \mathbb{P}_1)$ and a Markov kernel $\mathbb{P}_{1,2}:\mathbb{R}^k\times\mathcal{B}^l\rightarrow\mathbb{R}$ on a measure space $(\mathbb{R}^l, \...
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How computing a marginalization?
Let $u \in \mathbb{R}^K$ under the constraint that $u$ is sparse, i.e., $\|u\|_0 \ll K$. Sparse Bayesian technique considers $U$ as a random vector with parameters $\sigma_U^2=\left(\sigma_{U_1}^2, \...
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Finding the marginal distributions of a Gaussian mixture model. Is it the same as the Gaussian distribution?
Does the marginals of mixtures of Gaussians follow the properties of Gaussian distribution and the definition of marginalization?
What I want to do is to obtain the marginal probability density ...
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Deriving marginal distribution from a joint distribution
Let the joint distribution of a random vector be $$f(x,y)=\begin{cases}1,\enspace 0<x<1, x<y<x+1&\\0, \enspace {\rm otherwise}\end{cases}$$
The marginal pdf of $X$ is $Uni(0,1)$, but ...
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Can we pull back a measure with additional information on one coordinate?
Let $\mu$ be a finite measure on a product sigma algebra $(\Sigma_1\times\Sigma_2, \mathcal A_1\otimes\mathcal A_2)$, let $f:\Sigma'_2\to\Sigma_2$ be a measurable function from another measurable ...
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Radon-Nikodym derivative with respect to product of marginal measures
Let $\mu$ be a (finite if necessary) measure on the product $\sigma$-algebra $\mathcal A_1 \otimes \mathcal A_2$ of two measurable spaces $(\Sigma_1,\mathcal A_1)$, $(\Sigma_2, \mathcal A_2)$.
The ...
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Normal distribution for $Z=X+Y$ where $X,Y$ are both normally distributed
I need to find the probability distribution for $Z = X+Y$ where $X \sim \mathcal{N}(x_0,\sigma_x^2)$ and $Y \sim \mathcal{N}(y_0,\sigma_y^2)$. $X$ and $Y$ are independent.
In order to do this, we use ...
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Determining independence of random variables from joint pdf
I am working on the problem below and have a question about the note that is provided.
I know that independent random variables must satisfy the following;
$$
f_{XY}(x,y) = f_{_X}(x)f_{_Y}(y)
$$
...
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Doubly stochastic measures with decomposed marginal into convex combination of two other measures
Let $X,\mathcal{B}_X,\mu$ and $Y,\mathcal{B}_Y,\nu$ be probability measures on Borel-$\sigma$-algebra.
Let $\mu_i$, for $i=1,2$ be two distinct probability measures (and different from $\mu$) on $(X,\...
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Quantile function, bivariate joint density.
Consider random variables $X,Y$ with joint density function
$$
f(x, y)=\frac{1}{2 \pi}\left(1+x^2+y^2\right)^{-3 / 2}
$$
I want to find the quantile function of $|Y|$. I have learned how to find the ...
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When are S and T uncorrelated based on the marginal distributions of X and Y? With S = X - Y and T = X + Y
Given two random variables $X$ and $Y$, with $S = X - Y$ and $T = X + Y$.
Under what constraint on the marginal distributions of $X$ and $Y$ are $S$ and $T$ uncorrelated.
I know that $S$ and $T$ are ...
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how to derive a marginal PMF from this cumbersome joint PMF?
I've been learning probability theory recently and got stuck with this problem: knowing that the joint PMF of a 2 dimensional random variable $(X,Y)$ :
\begin{equation}P(X=n,Y=m)=\frac{\lambda^np^m(1-...
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Compute marginal density given conditional density
Given a continuous random variable $X$ with conditional pdfs $$f_{X=x|Y=0}\qquad \text{and}\qquad f_{X=x|Y=1}$$ and the probability of a discrete random variable $Y$ as $P(Y=0)=0.1,P(Y=1)=0.9$, how ...
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Marginal distribution from joint distribution
I have the joint distribution of two random variables $x,\theta$
$$h(x,\theta)=\frac{1}{2\pi\sigma}\exp\left\{-\frac{1}{2}\left[(x-\theta)^2+\frac{\theta^2}{\sigma^2}\right]\right\}$$
To find the ...
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Calculating marginal mean, variance, and pmf
Suppose that $Y|X \sim\operatorname{Poisson}(cX)$ where $c>0$ is a constant and $X\sim\operatorname{Exp}(1)$
(a) Find the marginal mean and variance of $Y$.
(b) Find the marginal pmf of $Y$.
For ...
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Distribution of $J$ if $K \sim \operatorname{Poisson}(\mu)$ and $J\mid K = k \sim \operatorname{Bn}(k,p)$.
My working so far:
$$
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
$$P_S(x) =...
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If $\pi\in P(X^2\times X^2)$ has marginals $\delta_x\times\delta_y$ and $\delta_x\times\nu$, then $\pi=\delta_x\times\delta_y\times\delta_x\times\nu$?
Let $X = \mathbb{R}$ and suppose that the probability measure $\pi\in P(X^2\times X^2)$ has the marginals $\delta_x\times\delta_y$ and $\delta_x\times\nu$, where $x, y \in X$ and $\nu$ is a ...
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Is there a marginal probability measure like a marginal pdf?
Forgive my notation! I don't know much about measure theoretic probability theory.
From undergrad probability I learned that a marginal density can be obtained from a joint density function using
$$f(...
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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
$$f(x,y)=
\begin{cases}
\frac{2e^{-2x}}{x} , & 0\le x \lt \infty \ , \ 0\le y \le x \\
0\quad , & \text{otherwise}
\end{cases}$$
...
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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 $g=0.$ ...
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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 $...
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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?
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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 ...
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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 ...
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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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Joint laws and marginal laws
We have 10 marbles enumerated from 1 to 10, and two boxes $B_1,B_2$. Marbles are inserted randomly in boxes. Let $X$ be a random variable counting the number of marble in $B_1$ and $Y$ a random ...
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Using general bivariate gaussian to extract marginal PDF from given bivariate PDF
I had a homework question to find the marginal probability density functions, $p_X(x)$ and $p_Y(y)$, given a join probability density function $p_{XY}(x,y)$.
I have solved the problem by integration i....
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PDF of a linear combination using joint PDF
Let $\vec{X}=(X_1,X_2)^T$ (no info if dependent) have a given probability density function $f_{\vec{X}}(x_1,x_2)=\begin{cases}
6x_1, \hspace{2mm} x_1+x_2<1, \hspace{2mm} x_1,x_2>0,\...
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Support for marginals of joint PDF
I have two joint PDF-s and three questions below.
I have a random vector' $(U,V)'$ and it's PDF. I need to find the marginal PDF-s for both $U$ and $V$. I tried deploying the idea from this anwer, but ...
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Conditional expectation of a bivariate function?
Let $Y = f(X, W)$ for two discrete random variables $X$ and $W$. If I know $P(Y = y |X=x, W)$, how can I get
$$E[Y | X=x]$$.
Is the following correct: $$E[Y|X=x] = \sum _w P(X=x, W=w) * f(x,w)$$
And ...
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Marginal distribution of a uniform distribution on a unit disk
I am trying to find the marginal pdf of a uniform distribution on a unit disk.
Let $f_{XY}=\frac{1}{\pi}$, where $X^2+Y^2\leq1$
Here's my attempt:
$$f_X(x)=\int^{\sqrt{1-x^2}}_{-\sqrt{1-x^2}}\frac{1}{\...
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Marginal probability of Y at a point
A car vendor sells type A and type B cars. Consider that X and Y
represent the types A and B of cars being sold (respectively).
Consider the Random Pair (X, Y) to have probability function according ...
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