Marginal probability distributions arise from joint probability measures on product spaces. The marginal distributions are the push-forward measures induced by the coordinate projections.
Questions tagged [marginal-distribution]
250 questions
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Difficult integral for a marginal distribution
I am trying to derive a marginal probability distribution for $y$, and failed, having tried all methods to solve the following integral:
$$
\operatorname{p}\left(y\right)…
Nero
- 3,779
7
votes
2 answers
Uniform distribution on unit disk
Let $(X, Y)$ be a random point chosen according to the uniform distribution in the disk of radius 1 centered at the origin. Compute the densities of $X$ and of $Y$.
I know that the joint density of $X$ and $Y$ is $\frac{1}{\pi}$ since when we…
DHH
- 433
6
votes
1 answer
Computing the finite-dimensional marginal distributions of Brownian Bridge
I'm working through Le Gall's Brownian Motion, Martingales, and Stochastic Calculus, and I'm struggling on an exercise. The question concerns computing the finite-dimensional marginal distributions of a Brownian bridge. In particular, let $B_{t}$ be…
Alex Lapanowski
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Joint distribution from marginals
I have a question about a joint distribution calculated in a paper I am reading.
There are three random variable a, b and c such that $$ a,b,c \in \{+1,-1\} $$ and then the joint distribution is given by:
$$ p(a,b,c) = \frac{1}{8}(1 + aE_{A} +…
Pegi
- 540
5
votes
1 answer
Does weak convergence of measures preserve independence of marginals?
Let $X^n = (X^n_1, \dots, X^n_d) ~ q^n$ be a $d$-dimensional random variable, where all the components are independent. That is, $X_i \perp X_j$ for $i\neq j$, and $$q^n(X) = \prod_{i=1}^d q^n_i(X^n_i).$$
If the sequence of measures $q^n$ converges…
blue_egg
- 2,333
5
votes
1 answer
difference of Bayesian inference using marginal and conditional distribution of multinomial model.
Say, there are 3 categories being selected with probability $\theta_i$ , $i=1,2,3$. After $n$ independent multinomial trails, we observe say $n_i$ outcomes of each $i$ category.
Then, someone told me that actually we can know $\theta_2=0.1$ for…
weidade3721
- 145
4
votes
1 answer
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 function of each $x_j$, for all $j=1,\ldots,n$, where…
Naomi
- 155
4
votes
1 answer
How can I get a marginal PDF from a joint PDF (probability density function)?
Let X and Y be random variables with a joint probability density function (joint PDF) given by
$ f_{X,Y}(x,y) \quad=\quad \begin{cases} \frac{c}{1+x^2+y^2} & \text{ if } x^2+y^2<1\,, \\ 0 & \text{ otherwise,} \end{cases} $
where the positive…
Jena Rayner
- 91
4
votes
2 answers
Property of distributions over R x R with identical marginal distributions
Let $D_1$ and $D_2$ be probability distributions on $\mathbb{R} \times \mathbb{R}$ with identical marginal distributions (i.e. the distribution of the first component of $D_1$ is the same as the distribution of the first component of $D_2$, and…
Eric Neyman
- 311
4
votes
1 answer
Marginalizing by sampling from the joint distribution
For two random variables $x$ and $y$, if I can sample from the joint distribution $p(x, y)$, I can obtain samples from the marginal $p(x)$ by sampling from the joint distribution and ignoring the values of $y$. I want to make a formal argument for…
cheersmate
- 143
4
votes
1 answer
Uniform distribution on the unit circle
Determine the probability density function on the unit circle $U:=\{(x,y)\in\mathbb{R}^2:x^2+y^2 =1\}$ with respect to the Lebesgue measure $\lambda$. Calculate the marginal distributions $f_X$ and $f_Y$ if the random positions on the unit circle in…
Analysis801
- 589
4
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1 answer
Where does the term "marginal" in "marginal probability" or "marginal distribution" come from?
Where does the term "marginal" in "marginal probability" or "marginal distribution" come from?
Daniel
- 589
3
votes
1 answer
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
$$
\begin{bmatrix}
0 & 0.5 & 0.5 \\
0.1 & 0 & 0.9 \\
0.8 & 0.2 & 0
\end{bmatrix}
$$
Determine whether the Markov chain has a…
Mohammed
- 109
3
votes
1 answer
Deriving marginal distribution from a joint distribution
Let the joint distribution of a random vector be $$f(x,y)=\begin{cases}1,\enspace 0
In the blind
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3
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1 answer
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 marginal measure of $\mu$ on $\mathcal A_i$ for…
Michael
- 375