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Questions tagged [marginal-distribution]

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I need to find the marginal distribution of Y from the following distributions

$f_X(x) = \frac{1}{2}e^{\frac{-x}{2}}$ and $f_{Y|X}(y|x) = I_{[0;x^2]}$ (Uniform continuous from $0$ to $x^2$). I tried finding the joint distribution by using $f(X,Y) = f(Y|X) * f(X)$ and then ...
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50 views

I'm trying to find a marginal distribution for a function and I need to solve the integral bellow.

I have arrived at the following integral: $$\int_y^\infty \frac{e^{-x/2}}{2x^2}$$ The limits I'm not so certain, but the function is correct. I have tried integration by parts but I arrived at a more ...
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1answer
46 views

What is better way to estimate marginal density?

Let there be the random variables $X$ and $Y$. We have a sample of $X$'s and Y's together. To most accurately estimate the density $f_X$, would we... 1) Ignore the Y's and estimate $f_X$ only from ...
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Is there any connection between marginal distributions and derivatives?

I know that "marginal" in "marginal distribution" is due to the fact that the marginal distributions of the discrete variables $X$ and $Y$ appear on the margin of the table used to express the ...
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Process for marginal distribution in multidimensional SDE using conditional expectations

Question Is it true in general that if you have a multidimensional process: $$ \begin{align} \mathrm{d}X_t &= \mu(X_t,Y_t)\,\mathrm{d}t + \sigma(X_t,Y_t)\,\mathrm{d}V_t \\ \mathrm{d}Y_t &= \...
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1answer
15 views

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 ...
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1answer
23 views

DEMONSTRATION FINITE-SAMPLE PROPERTIES OF LEAST SQUARES $\frac{(N-k)S^2}{\sigma^2}\sim\chi^2[n-K]$

Im a Student of Economics, and I have a concern. In the solution of $\frac{(n-K)S^2}{\sigma^2}\sim\chi^2[n-K]$ How can I show that if the matrix is ​​symmetric and idempotent between $(I-H)=|| (I-...
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2answers
43 views

Knowing the distributions of $X$ and $Y$, find the distribution of $Z=XY^2$

I know how to approach this kind of exercises, but I tried several times to calculate correctly the integrals, but I am doing some mistakes. So, let $X$ and $Y$ two independent variables with ...
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11 views

Computing the marginal of a complex Gaussian distribution

I have the following complex Gaussian conditional distribution \begin{align} p(x | \mu, \phi, \sigma) = \frac{1}{\pi^{\frac{1}{2}} \cdot \sigma } \exp \bigg( -\frac{1}{\sigma^{2}} \Big( \overline{x} -...
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1answer
19 views

Given the pdf of the random vector $(X,Y)$ find the pdf of $Y$

Let us consider the random vector $(X,Y)$ with pdf given by $$f(x,y)=\frac{5}{2}e^{-x-2y} \quad for \quad 0<x<+\infty, 0<y<2x.$$ Find the pdf of $X$ and the pdf of $Y$. I calculate the ...
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43 views

Find the marginal pdf of $X$ and $Y$

Suppose that $(U,V)$ has the following joint pdf: $$f_{U,V}(u,v)=\exp(-\theta u-\theta^{-1}v)$$ , where $u\geq0$, $v\geq0$, $\theta>0$. Define $X=UV$ and $Y=U/V$. Find the marginal pdf of $X$ and $...
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29 views

What does this property of marginal CDFs mean?

I am studying Join probability distributions and random variables. I've come across joint distribution functions (or cumulative distribution functions), and the following property: $F_X(x) = F_{XY}(x,...
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1answer
38 views

Problem to calculate a marginal function in probability

I have a problem in probability. I have $f(x, y) = \frac 14 \cos(y) $, if $x$ is between $0$ and $\pi$, and if$ y$ is between $-\frac x2$ and $\frac x2$. I have to calculate $f(y)$. I calculated ...
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Phase marginal for multivariate complex Gaussian density

The following is a cross-post from stats.stackexchange, which I am including here since it is mostly about a hard integral. Suppose $z$ is a random variable taking values in $\mathbb{C}^n$ and ...
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1answer
11 views

Is marginalization a sum of conditioned observations?

I have the following probability model: the joint probability p(A,B,D) is to be evaluated. I think there are 2 ways how to think about the problem: Marginalization over C of p(A,B,C,D) ignoring the ...
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1answer
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How do you marginalize in graphical model elimination?

I'm reading Michael I. Jordan's book on probabilistic graphical models, and I don't understand the elimination algorithm presented in chapter 3. To narrow the question down, consider page 6. In ...
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How does marginalizing over one variable affect independencies in the distribution?

I was requested to find a general algorithm which, given a Bayesian network graph $\mathcal{G}$ over a set of random variables $\mathcal{X}$ and a node to remove $X\in \mathcal{X}$, builds a new ...
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1answer
34 views

How to find limit of integration for marginal densities after transformation?

Here is the problem and solution for part b, which I need help with : How do you get the highlighted limits of integration? Four equations I have are: $u = xy$ $v = x/y$ $x = \sqrt{uv}$ $y = \...
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1answer
15 views

how to find Marginal probability function for Piecewise joint probability density

Given the following joint density function: $$ \displaystyle f(x,y) = \begin{cases} 0.5 & \text{if $0 \leq x < \frac{1}{3}, \> \frac{1}{3} \leq y < 1, $} \\ 0.75 & \text{if $\frac{...
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27 views

Marginalization of squared probability distribution

Suppose we have a probability distribution $P(x_1, x_2)$ and its transformation $\frac{1}{C} P^2(x_1, x_2)$ with some appropriate normalization constant $C$. Does the following hold: $$ \int_{-\infty}...
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1answer
26 views

What does it mean by Marginal Probability?

Suppose, we have the following distribution for a number of subscribers of HBO Network: \begin{matrix} \text{TV-Show/Sex} & \text{Male} & \text{Female} & \text{Sum}\\ \text{Game-of-...
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25 views

How to calculate marginal utility?

I am having some difficulties understanding the calculations of marginal utility. On this problem 𝑈(𝑃,𝑀)= square root of PM I know that MRS is equal to MUp/MUm but i do not understand how its ...
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23 views

Joint to marginal PDF, bounds of integration? (Bertsekas, Tsitsiklis, Question 3.4.15)

I am working on Question 3.4.15 from the second edition of Introduction to probability by Bertsekas and Tsitsiklis: "A point is chosen at random (according to a uniform PDF) within a semicircle ...
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29 views

Computation of marginal distribution for uncertainty quantification of dependent variables

In a few words, I have some dynamics with uncertainties in the initial conditions. I am using the Liouville equation and the method of characteristics to propagate in time the distribution of these ...
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30 views

Why joint distribution exercise goes wrong

Could anyone tell me what went wrong please. Don`t understand. I have an exercise: $$ f(x)=3x \\ 0 < y < x < 1\\ Find\\ P(X > \frac12 | Y > \frac13) $$ I found that: $$ P(X > \frac12 ...
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1answer
185 views

Given the joint density function of $X$ and $Y$, find the probability density function of $Z = XY$

The joint density function of $X$ and $Y$ is given by $f(x,y) = xe^{-x(y+1)}$, $x > 0, y > 0$. (a) Find the conditional density of $X$, given $Y = y$, and that of $Y$, given $X = x$. (b) Find ...
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Random vector and marginal distributions: integration limits and support related problems

Assume the random vector $(X,Y)$ has joint probability density function given by $$f_{X,Y}(x,y) = \begin{cases} kx(x-y) & \text{if}\,\,0 < x < 2\,\,\text{and}\,\,-x < y <x\\\\ 0 & \...
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24 views

Eigenvalues and -vectors of the marginal distribution

Can the eigendecomposition of a marginal distribution ($\boldsymbol\Sigma'V'=V'\Lambda'$ with $\boldsymbol\Sigma'$ a submatrix of $\boldsymbol\Sigma$) be easily derived from the eigenvectors and ...
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39 views

Upper bound for differences between two expectations

$f : \mathbb R^n \rightarrow \mathbb R$. Is there a good upper bound for the following difference? \begin{equation*} \big| \mathbb E_{(x_1, \ldots, x_n) \sim \nu} f(x_1, \ldots, x_n) - \mathbb E_{(...
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1answer
43 views

Relationship between conditional expectation and marginal join?

I have seen in book (Statistical Rethinking) this equation: $$ Pr(w) = E(Pr(w|p)) = \int Pr(w|p)Pr(p)dp $$ Where $$ Pr(w, p) $$ is join probably density function. Can somebody explain me the equality ...
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1answer
61 views

Find the density function $X+Y$, where $X$ and $Y$ have a joint density $f(x,y)=\lambda^2\exp(-\lambda y)$.

Find the density function $X+Y$, where $X$ and $Y$ have a joint density $f(x,y)=\lambda^2e^{-\lambda y}$ for $0\leq x \leq y, \lambda >0 $. I used the Density transformation theorem to find the ...
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1answer
21 views

finding marginal distribution - how to determine the limits of integration

This exercise comes from Rice 3.12: let $f_{XY}(x,y)=c(x^2-y^2)e^{-x}, 0\leq x <\infty, -x \leq y \leq x$ b) find the marginal densities I have found thatc $c=\frac18$ and also that $f_X(x)=\...
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Confirmation: getting joint cdf and marginals from joint pdf when region is 0<x<y

$h(x,y)= e^{-y}$ if $0<x<y<\infty$ and $=0 $ otherwise. To get the joint cdf $H(x,y)$, that is just found by integrating $h(x,y)$ over the region $\{(x,y):x \leq x_0, y \leq y_0, 0<x<y\}...
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2answers
23 views

What is the domain of my marginal PDF function

Consider the joint pdf of $(X, Y)$ given by $$f(x, y) = \begin{cases} 25x^{4}y^{4} & \text{ if } |x| \leq y, 0 < y < 1 \\ 0, & \text{ otherwise.} \end{cases} $$ Then, $$f_{X}(x) = \...
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38 views

Are the random variables $X$ and $Y$ independent given the law of $(X,Y)$ on $\mathbb{R}^2?$

Let $(X,Y)$ be a random vector whose density $f$ is given as follows: $$f(x,y) = \frac{\sqrt{3}}{2\pi}\exp\left(\frac{-1}{2}(4x^2+2xy+y^2)\right)$$ $\forall (x,y)\in \mathbb{R}^2.$ With this ...
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1answer
82 views

find $c$ in $f(x; y) = c(x^2 − y^2)e^{−x}$; $x > 0$; $−x < y < x$

find $c$ in $f(x; y) = c(x^2 − y^2)e^{−x}$; $x > 0$; $−x < y < x$ I know I have to take the integral such that : $$ \int \int f(x; y) = c(x^2 − y^2)e^{−x} dx dx = 1 $$ but I have troubled ...
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1answer
54 views

Computing the marginal probability distribution function for $f(x, y) = 24xy$

Suppose I wanted to compute $f_{X}(x)$ or $f_{Y}(y)$ for $f(x, y) = 24xy$ where $0 < x < 1 $, $0 < y < 1$ and $0 < x + y < 1$. I'm not sure about how to deal with the last ...
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Finding the joint density of two independent exponential distributions [duplicate]

Let $X_1,X_2$ be independent random variables each having a exponential distribution with mean $λ = 1$. (a) Find the joint density of $Y_1 = X_1$ and $Y_2 = X_1 + X_2$. (b) Get the marginal density ...
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39 views

density transformation theorem and marginalisation exercise using jacobian

Suppose that the joint probability density function of $X_1$ and $X_2$ is given by $$ f_{X_1X_2} = 6 \exp (-2x_1 - 3x_2) \text{ for } x_1>0, x_2>0 $$ we define $Y_1$ and $Y_2$ as follows: $$...
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1answer
23 views

Difficulty Understanding Solution to Marginal and Joint Density

I was able to solve part (a) using spherical coordinates. Part (b) is a completely different story. I'm not sure how the bounds were derived from the corresponding integral. Would someone on MSE be ...
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1answer
101 views

Find the marginal distribution given the mean and the covariance matrix

If we have a vector of normally distributed random variables $x^T = (x_1,x_2)$ with mean $\mathbf{\mu}^T = (10, 14)$ and a covariance matrix $$S_1 =\begin{bmatrix}13 & 12\\12 & 13\end{bmatrix}....
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1answer
22 views

Conditional densities and their sums

The question I'm working on boils down to finding the distribution of $Z_2$ if you know that $Z_1 \sim N(\mu_0,1)$ and $Z_2|Z_1 \sim N(Z_1,1)$ (more generally, we want to determine ${\mathrm{E}}[w_1\...
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1answer
44 views

Marginal distribiution of X and Y

"The joint density function of (X,Y) is given by: f(x,y)=2 for |x|<=0.5,|y|<=0.5 and xy>=0.Otherwise f(x,y) equals 0. Find the marginal distribitions of X and Y and check if they are independent....
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35 views

Joint PDF from Copula

I have two 1-d distributions and the correlation between them. I want to add them together, but to do this I'll need their joint distribution. I stumbled across copulas which seemed like a promising ...
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3answers
55 views

Find the density function of a random variable that depends on two other random variables with a given joint distribution.

For example, The joint density of $X$ and $Y$ is given by $$f(x, y) = \begin{cases}e^{-x-y}&\text{ if } 0<x<\infty, 0<y<\infty\\ 0 &\text{ otherwise}\end{cases}.$$ Find the ...
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2answers
335 views

Find joint CDF given a joint PDF

Let $X$ and $Y$ have a joint density function given by $$ f(x, y) = \begin{cases} 1, & \text{for } 0\leq x\leq2,\;\max(0,\,x-1)\leq y\leq \min(1,\,x), \\ 0, & \text{otherwise}. \end{cases} $$...
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1answer
64 views

Marginal distribution of sum of Poisson random variables from i=2 to n

This is a simple question but I'm not sure what to do when we're not summing the distributions from $i=1$ to $i=n$. Let's assume $X_1, \dots, X_n$ are i.i.d observations sampled from a Poisson ...
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1answer
21 views

Bayesian Statistics: Marginal Posterior

In a hierarchical model, the prior $\pi(\theta\mid\xi)$ for $\theta$ depends on hyperparameters $\xi$. In my lecture notes, the following is now given: $$ \pi(\xi \mid x) = { \pi(\theta,\xi\mid x)\...
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0answers
34 views

Marginal density, limits of integration

I have encountered another problem in my studies of probability. I add an image of the question as well as my solution, and below the solution given in the book. As far as I understand the concept of ...
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1answer
129 views

Marginal distribution of the unit disk

can you help me how to solve this task, I need also an explenation on formal definitions how to proceed with that question Task is on the picture