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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Marginalize Gaussian from exponential family distribution

Consider the marginal distribution $$ p(y|z)=\int p(y|f)p(f|z)df $$ where $p(y|f)$ a generic distribution in the exponential family and $p(f|z)$ is Gaussian and the parameters of both distributions ...
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Posets for conditional distributions over n variables

When considering distributions of up to n variables, enumerating the set of all joint and marginal distributions forms a poset conveniently arranged in an n-cube: poset of 0 to 3 variables. Thus there ...
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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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How do you calculate the determinant of the covariance matrix of a marginalized normal distribution given the Cholesky decomposition?

Calculating the probability density of a multivariate normal distribution involves calculating the determinant of the covariance matrix. Given that you know the Cholesky decomposition of the ...
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The upper bounds of integrals in the expression of marginal pdf of order statistic

The question comes from an expression of marginal pdf of order statistic in a text that I am reading. Two related pages are in the link below. https://imgur.com/a/yamubX7 (you need to scroll down for ...
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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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(marginal density) Consider the joint density of the random variables $X_1,X_2$...

Consider the joint density of the random variables $X_1,X_2$ $f(x_1,x_2)=ce^{−4x_1^2-\frac{x_2^2}{4}+\frac{x_1x_2}{2}}$ where $c \gt 0$ and $x_1,x_2 \in \mathbb R$. Determine the marginal density of $...
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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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Proper limit for such marginal distributions

Suppose we have a two-variable function f(x,y)=3x-y or even simpler f(x,y)=3 with a condition such as 0<=x<y<=1 . That is, y is always larger than x and x is always limited by y. What are the ...
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Finding bounds of integration from joint pdf for a marginal pdf of y

I have a joint function of X and Y such that \begin{equation} f_{X, Y}(x, y) = \begin{cases} \dfrac{5}{6x}, & \text{if}\ x \in \ [0, 2], \ y \in \ [-8x, 13x] \\ 0, & \text{else} \end{cases} \...
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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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Can we show that $\mathbb{P}_{\pi,\mu,q}(s,m,a)=\pi(s|\theta)\times\mu(m|s,\theta)\times q(a|m,s,\theta)$ is the joint pdf of $(s,m,a)$?

Suppose that we have the following probability measures $\pi:\Theta\to\Delta(S)$, where $\theta\in\Theta$ is the typical element of the state space (namely the state of the world) and $s\in S$ is a ...
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Single integral over bivariate normal distribution

I have $(X, Y)$ are bivariate normal with $\mu_X=log(160), \mu_Y =log(165),\sigma_x=0.05=\sigma_y$, and $\rho=0.5$. After a change of variables, I want the marginal density $f_U(u) = \int_{0}^\infty\...
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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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Determining the marginal density from joint probability

Given joint probability $$P(x,y) = \frac{1}{8\pi}e^{{-a(bx^2-cxy+dy^2)}}$$ where $a,b,c,d$ are rational fractions How would I go about determining the marginal probability of $p(x)$ and $p(y)$? I feel ...
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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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Understanding Bayesian Hierarchical Model in Practice

I have a Bayesian hierarchical model with datapoints $y_{ij}$ which are samples from distributions with parameters $\theta_j$. For each distribution parameter $\theta_j$, there are $n_j$ datapoints ...
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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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Defining joint distribution of two uniform variables where one is bounded by the other

I want to know how to define the joint distribution between two continuous uniform random variables. The first variable, X, is simply X ~ U(0, 1). The second variable, Y, can also vary between 0 and 1,...
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Mean of n-dimensional uniform unit-ball

Suppose I have an $n$-dimensional unit-ball: $$ \{(t_1,...,t_n) : t_1^2+...+t_n^2\le1\} $$ And suppose I have an $n$-dimensional random variable $(X_1,...,X_n)$ who distributes uniform in this unit-...
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Find the marginal densities of $X$ and $Y$ knowing $h(x,y)$

I have the distribution with joint density $h$ (with respect to the Lebesgue measure): $$h(x,y)=\frac{3}{2}y 1_{A}(x,y), \ \ A=\{(x,y) \in R^2|0<y, x^2+y^2<1\}$$ And then I have to find the ...
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Distribution of dot product of a random vector sampled from truncated normal distribution

I am trying to write down a PDF of $\mathbf{v} \cdot \mathbf{w}$ where $\mathbf{w}$ is sampled from an isotropic Gaussian truncated to $\mathbb{R}_{\geq0}^d$, and $\mathbf{v}$ is an unit vector. For $...
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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 ...
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How to get the marginal distribution by splitting the integral of one of two variables [closed]

I am trying to follow the answer to this question. Sorry to ask to read another post, but I don't understand why the integration of X is not just between the limits [0,1], where X has support. Instead ...
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Justification for the calculation of marginal distributions

My understanding of the concept of marginalising a joint probability distribution always came from the law of total probability, which states that for a finite or countably infinite partition $(B_n)_{...
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Formula of joint pdf

Let $\Omega$ be a sample space and $\mathbb{Q}=(\Omega, \mathcal{F}, \mathbb{P})$ standard probability space. Suppose that there exeist three different functions, say $$\mu:\Omega\to\Delta(X)$$ $$\tau:...
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Question regarding conditional and marginal distributions

I’ve stumbled upon this figure. You can see that the shaded area represents the conditional density function: $$f_{X_1|X_2}(x_1|x_2) = \frac{f(x_1,x_2)}{f_{X_2}(x_2)}$$ At the same time, this shaded ...
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Marginal Distribution Rewriting Equation

I have a question, for which I could not find an answer on the Internet. Given the formula for the marginal distribution: $$ p(x|c) = \int p(x|z,c)p(z|c)dz $$ Is it allowed to rewrite this equation to ...
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Finding the probability of a point bounded in a region from its joint pdf

Let $R$ be the region in $\mathbb{R}^2$ bounded by the three lines $y=2.1$, $y=x$, $y=−x$ and contains the point $(0,\frac{1}{2})$. Let $(X,Y)$ be the coordinate of a point chosen uniformly at random ...
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Same marginals and covariance but different joint distribution?

The question and answer in: If marginal probabilities equal, can we say anything about joint distribution? gives an example such that one can have two different joint distribution from the same ...
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Approximating the marginal of nested Wishart distributions

Consider that you have the following joint distribution over two matrices $A$ and $B$ both of which are of size $p \times p$: $$ B \sim \mathcal{W}_p(S, n) \quad \quad A|B \sim \mathcal{W}_p(B, n)$$ ...
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Calculate marginal pdf of $Y$ when marginal pdf of $X$ and conditional distribution of $Y$ is given

The Question So I have tried the formula $f_{X,Y}(x,y)=f_{Y\mid X}(y\mid x)\cdot f_{X}(x)$ but I am not able to figure out the bounds and I am not sure what to do after the bounds are found Please do ...
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analytic multivariate Lorenz surfaces

I am given the projections of a multivariate probability distribution and asked to recover the multivariate distribution. How do I derive the copula? How do I use the copula to define the joint ...
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How to show the Joint CDF

I have a such example in my text book Let Y and Z be two IID random variables each with CDF, $F(\cdot)$. Let $X_1:=min(Y,Z)$ and $X_2:=max(Y,Z)$ with marginals $F_1(\cdot)$ and $F_2(\cdot)$, ...
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Mixture models vs marginal distributions

Maybe I'm not understanding these concepts correctly, but it seems like the idea of a mixture model seems to be superfluous, since we can express any mixture of random variables as a multivariate ...
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Marginalization of intermediate variables $p(w|z, y) = \int_z p(w| x, z)p(z| x, y)dz$

I want to know if my statement and derivation of the marginalization of the "intermediate" variable is correct. The statement is as follows: Suppose a probabilistic factorization model $p(x,...
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Obtain $P(A,B,C)$ from $P(A,B)$ and $P(B,C)$ if $A$ and $C$ are independent

If I am given the data of the marginals $P(A,B)$ and $P(B,C)$ together with the promise that $A$ and $C$ are independent, i.e. $P(A,C)=P(A)P(C)$, then, is there a way to deduce the full distribution $...
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Is this question correct because if I find the PDF in terms of X and Y wouldn't that be the marginal PDF f XY(x,y)? [closed]

Let S = { (𝑥,𝑦,𝑧) ∈ R³ | (𝑥² + 𝑦² + 𝑧²) ≤ 1, 𝑧 ≥ 0 } for any lebesgue measurable subset A of S, let P(A) = (3÷(2π)) × (volume of A) let X (𝑥,𝑦,𝑧) = 𝑥 and Y(𝑥,𝑦,𝑧) = 𝑦, ∀(𝑥,𝑦,𝑧) ∈ S. ...
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Computing Tables of Random Vectors

I have been trying to solve problems on random vectors, given their joint pmf table computing their expectation, variance and filling in the joint probability. I am finding difficult the filing joint ...
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Understanding this derivation of a conditional distribution for an autoregressive process

In a current paper I am reading, the following section shows up as a side note. I am not sure I understand the derivation the authors propose here: Consider the following auto-regressive process: $$...
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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)...
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For given Joint pdf, Find the value of $c$

My Working: So I have been practicing for my upcoming exam. This is from one of my past exam papers, and I am unable to solve it. The thing that confuses me is that the given pdf $f_X,_Y(x,k)$ ...
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