Questions tagged [marginal-probability]

Marginal probability arises from a joint probability measure on a product space. The marginal probability distributions are the push-forward measures induced by the coordinate projections. A marginal probability is the probability of a single cylinder-set event. This is contrast to joint probability or conditional probability, in which additional events are considered.

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33 views

Find the density function of $X+Y$ when $X,Y \sim_{i.i.d} U[0,1]$

I'm using the transformation of R.V and calculating marginal density approach and it confuses me how there's two cases. In my working I let $U=X+Y, V=Y$ $f_{U,V} = f_{X,Y}|J|=f_{X,Y}=1$ from $0<x&...
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14 views

Splitting of marginal PDF

If we have the function: $$f_{X,Y}\left(x,y\right)=\frac{3}{7}x,\:\text{ when }\:1\le x\le 2\:\wedge \:0\le y\le x$$ If I want to find the marginal function $f_Y\left(y\right)$. It should be: $$f_Y\...
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Transforming uniform probability density function over unit disc from polar coordinates to cartesian coordinates

I need to solve the following exercise and I am not fully sure about my approach as the results seem odd and therefore would like some advise. Given are uniform random distributions of an angle $\...
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1answer
28 views

Determining the Distribution of an uncorrelated Random Variable by using Dirac's Delta

Choosing random variables $X \sim U(-1,1)$ and $Y = X^2$, it is possible to show that X and Y are uncorrelated, yet not independent. I was wondering what the probability distribution of $Y$ now looks ...
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Checking if jointly CRVs are independent via the marginal density

I'm trying to determine whether the following continuous random variables are independent or not given their joint probability density. $f_{X,Y}(x,y) = 2(x+y)$ for $ 0\le y\le x\le 1$ I calculated the ...
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39 views

Marginalisation - Integrate out a variable

I calculated a joint density function of two non uniformly distributed variables. $$ f_{x,y}(x,y) = \frac{1}{2\pi}\cdot{1 \over \sqrt(x^2+y^2)}$$ I want to "integrate out" one of the ...
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21 views

Marginalization of conditional probabilities

An exam question asked to select all expressions that are equal to 1, given no independence assumptions. The solution stated that the following expression was not equal to 1, but I don't understand ...
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44 views

Interpretation of probability density greater than one

Good morning to everyone, I'm trying to figure out how to interpret a probability density greater than one, but I didn't find any explanation that I considered satisfactory... Let me formalize my ...
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1answer
67 views

Why do elliptical copula densities contain $x_1$ and $x_2$, but Archimedean copula densities contain $u_1$ and $u_2$?

$$c\left(u_{1}, u_{2}\right)=\frac{1}{\sqrt{1-\rho_{12}^{2}}} \exp \left\{-\frac{\rho_{12}^{2}\left(x_{1}^{2}+x_{2}^{2}\right)-2 \rho_{12} x_{1} x_{2}}{2\left(1-\rho_{12}^{2}\right)}\right\}$$ is the ...
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How to get the marginal density function of dependent random variables

I am unsure how to get the marginal probability density function if the two random variables are dependent. In general for given $f_{XY}(x,y)$ we get $$f_X(x)=\int_{-\infty}^{+\infty}f_{X,Y}(x,y)dy\...
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33 views

Is always true for the formula of marginal density function from joint distribution?

This is the formula to help you get the marginal pdf from a joint distribution: $$f(x)= \int_{}^{} f(x,y) \, dy\\$$ $$f(y)= \int_{}^{} f(x,y) \, dx\\$$ where the first integral is over all points in ...
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29 views

Integral of the square of a non-standard gaussian pdf [duplicate]

What is the integral of the square of a non-standard univariate gaussian distribution? The following is what I have so far, is it right? $\int_{-\infty}^{\infty} p^2(x)dx$, where $p(x)\sim N(\mu, \...
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19 views

Implications of independence for two dichotomous random variables

I am trying to understand the relationship among the following: $P(A | B)$, $P(A | \lnot B)$, and $P(A)$ A is independent of B iff: $P(A | B) = P(A)$ But, if A is independent of B, then does this ...
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23 views

What kind of probability do we have?

Among $100$ products we do qualitative test. Since we have already tested $20$, we test then the rest. The probability to have confident products at the second test, having tested the first ones, is ...
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Given a coupling $\pi(\mu,\nu)$, show that $E_\mu f- E_\nu f= E_\pi [f(X) - f(Y)]$

In the lecture notes by for High-Dimensional Probability by Handel, the following is affirmed: Let $\mu$ and $\nu$ be probability measures, then $$\mathcal C(\mu,\nu) = \{ \text{Law} (X,Y) : X\sim \mu,...
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42 views

What happen if a probability function does not converge?

Find the marginal distribution for $y_2$ given the following PDF $$f(y_1,y_2)= \begin{cases} 3y_1, & \text{if } 0\leq y_2\leq y_1 \\ 0, & \text{elsewhere} \end{cases} $$ But when I try to ...
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1answer
34 views

Showing function is not independent

I need to show that this is not independent, I found that $f(x,y) = 1$ with the given restrictions on $x$ and $y$, but now to find the marginal density functions, would the integral for finding $f_x(x)...
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23 views

What is the probability that a sample belongs to a particular random variable?

Edited for clarity. Let $X$ be a continuous random variable with a known probability distribution. Let $\mathbf{v}$ be a vector in $\mathbb{R}^n$. What is the probability that $\mathbf{v}$ is a sample ...
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1answer
24 views

Independence between fractional parts of consecutive sums of independent uniforms

Let $X_1,X_2$ be independent $\text{Uniform}(0,1)$ random variables. Define $U_1 = X_1 - \lfloor X_1 \rfloor$ and $U_2 = X_1 + X_2 - \lfloor X_1 + X_2 \rfloor$ where $\lfloor a \rfloor$ is the largest ...
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Concept about marginal probability $p(y)$to conditional probability $p(y|x)$ transformation?

I have a function like the following, $p\left( y \right) = \int\limits_x {\int\limits_z {(Q({x^2} + y) + yz + z)dxdz} } $ Where, $Q(x) = \frac{1}{{2\pi }}\int\limits_x^\infty {{e^{ - \frac{{{t^2}}}{2}...
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Marginalization : transformation formula between a $Y$ random variable and a function $X = g(Y)$ with $X$ another random variable

From the following link : Marginal distribution, I don't undertand the last formula and reasoning at the end of this quote : " A marginal probability can always be written as an expected value: $...
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1answer
28 views

Joint probability from marginal and relation between variables

I would like to know whether it is possible to obtain the joint density function $p(x,y)$ if I know one marginal, which is Gaussian $p(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp(-\frac{1}{2} (\frac{x - \mu}...
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97 views

Find the marginal probability function of X and Y

Problem $X$ and $Y$ random variables, the common probability density function of $X$ and $Y$ is given as follows: $$f(x,y)= \begin{cases} 2e^{-x-y},&\textrm{when } x\geqslant y\geqslant 0\\ 0\;,&...
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22 views

Continues joint distribution marginal PDF calculation

I have a joint function f(x,y) and i have two regions as stated on the above graph but i don't know if i am going to calculate Marginal PDF of X and Y differently on both regions and because ...
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135 views

Understanding sum rule for marginal probability

If $p(x,y)$ is the joint distribution of two discrete random variables $x, y$. The sum rule states that: $$p(x) = \sum_{y \in T} p(x,y)$$ Where $T$ are that states of the target space of random ...
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331 views

Joint Probability Mass Function/ Marginal Probability Mass Function

Q. A coin is biased so that heads appears with probability 2/3 and tails with probability 1/3. This coin is tossed three times. If X denotes the number of heads occurring and Y denotes the number of ...
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58 views

Suppose $X$ and $Y$ are discrete random variable with joint p.f. Find the marginal p.f. of $X$ and $Y$

The joint probability function of x and y is: $$f(x,y)=\frac{e^{-2}}{x!(y-x)!}$$ where $$x = 0, 1, ..., y; y= 0, 1, ...$$ My solution to this problem: The marginal p.f. of x is: $$f_x(x)=\sum_{y=0}...
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55 views

Marginal probability density function for a Joint Variable with two upper bounds

My joint variable function is $f(x,y) = \frac{2x+y}{36}$, $ 0 \leq y \leq x$, $x + 2y \leq 6$, zero otherwise. My question is when trying to find the marginal p.d.f. of X, how can I do that when y ...
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1answer
34 views

How can we know for use that $A$ and $B$ are mutually exclusive under conditonal probability?

Reading that "Sample question 4 (Conditional Probability): Given that $P(A) = 0.20, P(B) = 0.60, P(B|A) = 0.50,$ what is $(B∩A’)$ ? $(A’∩B’)$?" from datasciencecentral.com When we are just ...
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54 views

Two dimensional random variables and conditional pmf if $f(x)=24x^2, 0<x<\frac{1}{2} \text{ and} f(y|x=X)=\frac{y}{2x^2}, \text{ if} 0< y<2x$

My solution: $$f_Y=\int^{0.5}_{0.5y}12y dx=\left. 12xy \right|^{0.5}_{0.5y}=6y-6y^2$$ $$f(x|Y=y)=\frac{12y}{6y(1-y)}=\frac{2}{1-y}$$ In general, I think I don't understand what I have to do at all. ...
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1answer
98 views

A joint probability density function

Consider the joint probability density function $$ f_{X,Y}(x,y) = \begin{cases} \frac1y,& 0<x<y, \ \ 0<y<1\\ 0,& \text{otherwise.} \end{cases} $$ (a) Find the marginal ...
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1answer
22 views

From joint to marginal - Steps in-between unclear

I'm trying to understand the paper "Direct Uncertainty Prediction for Medical Second Opinions" [Raghu et. al., 2019] but I am missing some small intermediate steps to fully understand their equation. ...
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34 views

Continuous and discrete joint probability

i have attempted parts a ) b ) to which is in the link https://ibb.co/LQnztn0 however i cannot seem to understand parts c) and d ) my attempt to try part c) but then got stuck https://ibb.co/...
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69 views

Compact support of measure.

Let $\mu$ and $\nu$ two probabilites on $\mathbb{R}^{d}$ Let $\Pi(\mu,\nu)$ is the subset of the probability measure $\pi$ such as $$ \pi (A\times Y) =\mu(A) \text{ and } \pi(X\times B)=\nu(B) $$ ...
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1answer
107 views

Moment Generating Function of Y

f(x,y)=8xy for 0 What will be the MGF of Y? Do I need to insert the marginal density function for Y in the formula for MGF? Also, what will be the limits? Will it be from x to 1 or from 0 to ...
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30 views

Marginal Probability Density

if $A=1$, $y \sim N(1,\sigma^2)$ if $A=2$, $y \sim N(2,\sigma^2)$ $Pr(A=1)=0.5$ $Pr(A=2)=0.5$ Given $\sigma = 2$, I'm trying to find the formula for the marginal probability density function for ...
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48 views

marginal probability, probability,

A mosquito is trying to annoy you. At any given time, it can be found anywhere within a radius R from your center (assume that it can be found inside you). Use spherical coordinate system. i. ...
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3answers
250 views

Gaussian 2-D Mixture, mean mode median of marginals and 2-D

Let p(x1,x2) = $\dfrac {4}{10}\mathcal{N}\left( \begin{bmatrix} 10 \\ 2 \end{bmatrix},\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\right) + \dfrac {6}{10} \mathcal{N}\left( \begin{bmatrix} 0 ...
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How to compute U = X/Y of a pdf

I know that similar questions have been posted before but I don't understand how to solve them, especially because this very type of question is never explained in our textbook. Assume that we are ...
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35 views

Simple Derivation behind Joint Probability Statement

I've been looking at Michael Jordan's graphical model introduction at: https://www.cs.cmu.edu/~aarti/Class/10701/readings/graphical_model_Jordan.pdf page 8-9, where he does a derivation that is not ...
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96 views

Existence of Probability Measures With Given Marginals and moment constraints

let $\mu$ and $\nu$ be two probability measures on $\mathbb{R}^d$, my question under which conditions is there a coupling measure between $\mu$ and $\nu$ that satisfies some moment constraints. for ...
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78 views

Is P(Y = y) in the marginal probability formula always 1? Any special cases where P(Y = y) < 1?

The marginal probability equation follows: \begin{equation} \sum_{Y} P(X = x | Y = y)P(Y = y) \end{equation} Practically, it seems that most computations actually utilize the sum of the conditional ...
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1answer
64 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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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
278 views

Understanding marginalizing over many variables using joint probabilities.

If we have a joint probabiliy $p(x, y)$, where each are binary variables having $K$ values, we can compute the marginal probability $p(y)$ as: $$p(x, y) = p(y|x)p(x) \tag 1$$ $$p(y) = \sum_{x^{*}}p(...
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1answer
24 views

Definition of a Domain in Pan and Yang's article: “A survey of transfer learning”

In the article "A survey of transfer learning": https://ieeexplore.ieee.org/abstract/document/5288526 Pan and Yang give a good mathematical definition of transfer learning which seems to be widely ...
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1answer
32 views

Marginal density for joint pdf of multiple random variables

I have the joint pdf $f(x_1,x_2,x_3)=12x_2 \;\mathrm f \mathrm o \mathrm r \; 0<x_3<x_2<x_1<1,$ and $0$ elsewhere. I want to find the marginal pdf $f_{x_3}(x_3)$. I know that to do this ...
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1answer
341 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
120 views

How do I determine the domain of marginal density functions?

I have the following density function. $$f(x,y)=\frac{1}{32}*(10-3x^2-y)\quad,-1<x<1; 0<y<2$$ When calculating the marginal density for $x$, $f_x$, I get $$\int_{0}^2\frac{1}{32}*(10-3x^2-...
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1answer
54 views

Intuitively use marginalization

Is it always true that if you sum over some variable then you can "remove" that varaible in each expression of the product inside the sum? For example: $ \sum_x P(x, y)P(y | x)P(y | x, z) = P(y)P(y)P(...