Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [marginal-distribution]

The marginal distribution refers to the probability distribution of a subset of variables contained in a joint distribution.

The marginal distribution refers to the probability distribution of a subset of variables contained in a joint distribution. It is obtained by "summing over" all outcomes of the other variables in the joint distribution in case of discrete variables, and "integrating over" all outcomes of the other variable in case of continuous variables.

Thus, if $P(x_1,x_2,\ldots,x_n)$ represents a discrete joint distribution, the marginal distribution of $x_i$ is:

$$P(x_i) = \sum_{x_1,x_2,\ldots,x_{i-1},x_{i+1},\ldots x_n} P(x_1,x_2,\ldots,x_n)$$

The summation is over all possible outcomes of the indicated variables. Similarly, for the case of a continuous joint distribution:

$$P(x_i) = \int \int \ldots \int P(x_1,x_2,\ldots,x_n) dx_1 dx_2 \ldots dx_{i-1} dx_{x+1} \ldots dx_{n}$$

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What is a difference between random effects-, fixed effects- and marginal model?

I am trying to expand my knowledge of statistics. I come from a physical sciences background with a "recipe based" approach to statistical testing, where we say is it continuous, is it normally distributed -- OLS regression. In my reading I have…
N26
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Introductory reading on Copulas

For some time now, I have been looking for a good introductory reading on Copulas for my seminar. I am finding lots of material that talk about theoretical aspects, which is good, but before I move onto them I am looking to build a good intuitive…
NaN
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How to find marginal distribution from joint distribution with multi-variable dependence?

One of the problems in my textbook is posed as follows. A two-dimensional stochastic continuous vector has the following density function: $$ f_{X,Y}(x,y)= \begin{cases} 15xy^2 & \text{if 0 < x < 1 and 0 < y < x}\\ 0 &…
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Sampling from marginal distribution using conditional distribution?

I want to sample from a univariate density $f_X$ but I only know the relationship: $$f_X(x) = \int f_{X\vert Y}(x\vert y)f_Y(y) dy.$$ I want to avoid the use of MCMC (directly on the integral representation) and, since $f_{X\vert Y}(x\vert y)$ and…
Rod
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What are the differences between "Marginal Probability Distribution" and "Conditional Probability Distribution"?

While studying probability, I am kind of having difficulties in understanding marginal probability distribution and conditional probability distribution. To me, they look much the same and cannot find the clear concepts of differences in those two…
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Intuition behind the names 'partial' and 'marginal' correlations

Does anybody have an idea about why conditional correlation between 2 variables is called "partial" correlation and the simple correlation between them (so, when not conditioned on any other variable) is called "marginal" correlation? What is the…
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Maximum likelihood estimator of joint distribution given only marginal counts

Let $p_{x,y}$ be a joint distribution of two categorical variables $X,Y$, with $x,y\in\{1,\ldots,K\}$. Say $n$ samples were drawn from this distribution, but we are only given the marginal counts, namely for $j=1,\ldots,K$: $$ S_j =…
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Robust MCMC estimator of marginal likelihood?

I'm trying to compute the marginal likelihood for a statistical model by Monte Carlo methods: $$f(x) = \int f(x\mid\theta) \pi(\theta)\, d\theta$$ The likelihood is well behaved - smooth, log-concave - but high-dimensional. I've tried importance…
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Marginalization of conditional probability

I am working through these examples of computations on Bayesian networks and came across this claim (part of the last sample computation): $$ P(E=e|A=a) = \sum_{c \in C} P(E=e, C=c | A=a) $$ I am newly familiar with marginalization, but I thought…
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Updating a Bayes factor

A Bayes factor is defined in Bayesian testing of hypothesis and Bayesian model selection by the ratio of two marginal likelihoods: given an iid sample $(x_1,\ldots,x_n)$ and respective sampling densities $f_1(x|\theta)$ and $f_2(x|\eta)$, with…
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Is the maximum entropy distribution consistent with given marginal distributions the product distribution of the marginals?

There are generally many joint distributions $P(X_1 = x_1, X_2 = x_2, ..., X_n = x_n)$ consistent with a known set marginal distributions $f_i(x_i) = P(X_i = x_i)$. Of these joint distributions, is the product formed by taking the product of the…
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Problem calculating joint and marginal distribution of two uniform distributions

Suppose we have random variable $X_1$ distributed as $U[0,1]$ and $X_2$ distributed as $U[0,X_1]$, where $U[a,b]$ means uniform distribution in interval $[a,b]$. I was able to compute joint pdf of $(X_1,X_2)$ and marginal pdf of $X_1$. $$…
user1102
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How to calculate the standard error of the marginal effects in interactions (robust regression)?

what I am interested in learning is how to calculate the std error of the marginal effects of a X variable when it is part of an interaction, especially in robust regression. There are tipically two cases that interest me: when there is an…
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Marginal distribution of uniform distribution over sphere

Let $(x_1,…,x_n)$ be a random vector uniformly distributed on the $n$-dimensional unit sphere. Is there a closed form solution for the joint distribution of $P(x_1, x_2)$?
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What is the meaning of 'Marginal mean'?

My problem with understanding this expression might come from the fact that English is not my first language, but I don't understand why it's used in this way. The marginal mean is typically the mean of a group or subgroup's measures of a variable…
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