# Questions tagged [marginal-distribution]

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

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### Let random variables X and Y have the following joint density [closed]

Let random variables X and Y have the following joint density: fX,Y (x, y) = e^−y for 0<x<y; 0 otherwise; Find marginal density of X and Y, conditional density of X|(Y = y) and show that (Y − X)|...
1 vote
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### Finding the non-zero region of a marginal

Basic question but: what happens to the region on which a pdf is non-zero when a bivariate is integrated to get a marginal? The example I'm working on (course problem booklet for a mathematics BSc ...
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### Marginal posterior distribution of error variances

I have been working on Bayesian statistics recently and have came across the term called Marginal distribution of error variances. Though I understand what is a marginal distribution and that an error ...
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### Hierarchical model for uniform random variable

I am thinking about the following model: $$\theta \sim \mathcal{U}[c- \epsilon, c+\epsilon],\\ x \mid \theta\sim \mathcal{U}[\theta - \epsilon, \theta + \epsilon].$$ I want to derive the marginal ...
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### Expectation of a joint distribution vs product of marginal expectations?

I got into a conversation with a coworker, he was doing napkin math and showed that average purchase value and average conversion rate, together, give us the expected revenue. And my response was- not ...
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### Sampling from marginal densities using the joint

I have seen this approach link in different places. Hence, I suppose that the following method is correct. Let $f_{X,Y,Z}(x,y,z)=f_{X|Y,Z}(x|y,z)f_Y(y)f_Z(z)$ and I can easily sample from these 3 ...
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### quantile surface of a mulitvariate distribution made of multiplication of marginal distributions assuming independence

How to perform quantile regression in a more elegant fashion? As discussed above, quantSheets() can only deal with one explanatory variable for computing quantile ...
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### How can the marginal distribution be derived from conjugate Gaussians?

In An Introduction to Empirical Bayes Data Analysis by George Casella (1985), it is given that \begin{align} x|\theta &\sim N(\theta,\sigma^2) \\ \theta &\sim N(\mu,\tau^2) \end{align} and ...
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### How can we assume the models are exhaustive in Bayesian Model Averaging?

Bayesian model averaging is justified using the law of total probability which requires the the set of models that we average over to be exhaustive. Shouldn’t we prove that the set of models are ...
1 vote
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### Exponentiated Weibull-logarithmic Distribution

I'm trying to deduce the marginal cdf of $Y$ in Exponentiated Weibull-logarithmic Distribution from this paper: Exponentiated Weibull-logarithmic Distribution: Model, Properties and Applications In ...
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### Noob question about bayes rule denominator estimation

A known problem of Bayes rule is the intractability of the estimation of $p(D)$ given a multiparametric problem, since $p(D)$ is found by marginalizing the joint probability $p(D, \theta_{1..n})$ over ...
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### Approximating joint distribution from marginals and additional information

Consider a population generation question where we are trying to generate couples that conform to a local areas demographics. We know the age distribution for Partner 1, $x_1\sim D_1$, and for ...
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### Estimate a marginal (general) likelihood of best success time

In the figure below, I estimated two nonparametric pdf's: (x-axis is time in 24hour format, range between 8 to 20) green line: clock time of success phone calls (eg. at 4pm I called the customer, and ...
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### Finding the marginal distribution of log-normal random variable whose mean is dependent on a Gaussian random variable

My goal is to be able to integrate out the observation error of $\hat{X}$ in the set-up below, in order to compute the likelihood of $\frac{X}{\hat{X}}$ over all possible values of $\hat{X}$ : The ...
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### Intergrating product of multivariate normal and univariate normal to find marginal density [duplicate]

Suppose, $y_i|u_i\sim MN(X_i(t_i), \sigma_e^2I_{m_i})$ and $U\sim MN(0,I_p)$. Now how to find the marginal distribution of $f(y_i)=\int f(y_i|u_i)f(u_i)du_i$? Since one is multivariate normal and ...
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### Integrating product of multivariate normal and univariate normal to find marginal

Suppose, $y_i|u_i\sim MN(X_i(t_i), \sigma_e^2I_{m_i})$ and $u_i\sim N(0,1)$. Now how to find the marginal distribution of $f(y_i)=\int f(y_i|u_i)f(u_i)du_i$? Since one is multivariate normal and other ...
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### Generating an analytical copula for an example problem

I am currently doing research that requires me to understand dependence modeling. As a first step, I am reading An Introduction to Copulas. I am, stuck on the first example problem which I have re-...
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### Marginal and joint distributions of normal and student's

Is there an equivalence between two normal (student's) distributions. in the sense that, if my two variables (X,Y) are normal (student's) then their joint distribution is also normal (student's). I ...
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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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### How to derive marginal likelihood equation for linear regression as product of two Gaussians?

I understand that marginal-likelihood can be derived as answered here. Quoting the same proof from MATHEMATICS FOR MACHINE LEARNING book (9.3.5) Page 312, The same book mentions that we can derive ...
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