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Questions tagged [marginal-probability]

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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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How to simplify multiple events conditioned on a single event?

$\Pr(P_0(D_1\cup D_2\cup D_3)) = \Pr((P_0D_1)\cup(D_2D_3))$ $= \Pr(P_0D_1) + \Pr(P_0D_2 \cup P_0D_3) -\Pr(P_0D_1D_2D_3) $ $= \Pr(P_0D_1) + \Pr(P_0D_2) + \Pr(P_0D_3) - \Pr(P_0D_2D_3) - \Pr(...
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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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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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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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1answer
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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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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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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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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(...
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marginal probabilities, multivariate random variables

I want to solve the task below... However, I have a problem with the marginal probabilities not adding up to 1. what's wrong?
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Show that the marginal probability $P(X = x)$ is given by:

The random variables $X$ and $Y$ take integer values $x$ and $y$, both $≥ 1$, and such that $2x + y ≤ 2a$, where a is an integer greater than $1$. The joint probability within this region is given by: ...
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Continuous Conditional Probability Question

A system consisting of two components will continue to operate only as long as both components function. Suppose the joint pdf of the lifetimes (months) of the two components in a system is given by $...
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Limits of Integration When Finding Marginal PDF

My question is about the same exercise problem asked here. I had a different question though regarding it, and decided to ask my own question. More specifically, I keep seeming to have trouble ...
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Deriving marginal likelihood formula

The formula for marginal likelihood is the following: $ p(D | m) = \int P(D | \theta)p(\theta | m)d\theta $ But if I try to simplify the right-hand-side, how would I prove this equality $ = \int \...
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Independence of two disjoint sums of independent Random variables

Suppose in a probability space $(X,\Omega,\mu)$, let $X_1,X_2,...X_n$ be independent random variables. How do I prove that $X_1+X_2+...+X_{i-1}$ is independent of $X_i+...+X_n$ for some $1\leq i \leq ...
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limits of marginal probability of polar coordinate probability

I'm self-studying probability and came across a problem of finding the marginal probability $f_\theta$ of a pdf in polar coordinate $f_{r,\theta}$. To find the $f_r$, I know we do $\int^{2\pi}_{0} f_{...
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Notation of marginal probabilities

I found a (to me) strange notation concerning marginal probabilities I don’t understand. Unfortunately I will include picture of the notation. Does it mean x=2.5, shouldn’t it be 0 then? Do they mean ...
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Integration help for marginal probability density problem

Trying to understand the following integral used to find the marginal probability density function for continuous random variables x & y. I haven't taken a calculus class in 20+ years, so this is ...
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Marginal probability density of a free particle of mass m

I'm trying to solve this problem but first of all I can't find the relation between the probability density at time $t=0$ and the probability density at time $t$. Anyone can help me? This is the ...
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Conditional normality implies marginal normality or bivariate normality? [closed]

Let X and Y be two random variables. If we know that both Y|X and X|Y are normal, then can we conclude that (1) X and Y are normal respectively; (2) X and Y are bivariate normal? Thanks! Edited: ...
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Finding marginal distribution under change of variable of pdf

Question: Let $X$ and $Y$ be independent random variables, each with probability density function $$f(x) = \lambda e^{−\lambda x} $$for $x > 0$. Let $U = X − Y$ and $V = X + Y$ . Find the joint ...
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Joint distribution probability

Hi , I tried to resolve this question by taking the marginal pdf of $Y$ , then find it's expected value . But apparently this is not correct, they used the pdf of $f(x,y)$ in the solution and ...
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Is there any situation where the joint probability is equal to the marginal probability?

Is there any situation in that joint probability $p(x,y)$ equals to marginal probability $p(x)$? What is the interpretation of this situation?