# 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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### Is it possible to decompose the joint conditions?

Given three events $E$, $A$ and $B$, is it possbile to decompose the joint-conditional probability $P(E \vert A, B)$, as a expression in terms of non-joint-conditional probabilities, and marginal ...
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### Joint probability table not adding up to 1

I got this question asked in my exam today: Suppose there is a group of 9 students that belong to a university. 4 of them study economics, 3 of them study business and the rest study engineer. Suppose ...
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### I can't seem to understand marginal probabilities.

In the book Pattern Recognition and Machine Learning, CM Bishop has elaborated on calculating marginal, conditional and joint probabilities by creating a table with rows and columns being outcomes of ...
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### Prove $X$ and $Y$ with $X \sim \text{Unif}(0, 1)$ and $Y = X$ have no joint probability density function

Prove that the two random variables $X$ and $Y$ with $X \sim \text{Unif}(0, 1)$ and $Y = X$ have no joint probability density function (PDF), while each margin has a PDF. Here is my progress, please ...
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### Can marginals determine all probabilities

suppose we are given $n$ binary random variables, $X_1,\dots,X_n$. A probability distribution $P$ assigns all elementary events a probability; $$P(X_1=x_1\&\ldots \& X_n=x_n)\in[0,1].$$ A ...
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### Probability density of two random variables using characteristic function

I've been trying to solve the following question : $X$ and $Y$ are two real random variables with a probability density of : $$f(x,y) = e^{-y} *\mathscr{1}_{0<x<y}(x,y)$$ where $\mathscr{1}$ is ...
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### What does it mean to divide by a pdf

We know the formula, f(A|B) = f(A,B)/f(B). In my mind the left hand side makes sense since we are basically taking a 2D distribution to a 1D distribution(well it does not necessarily integrate to one ...
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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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### 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 Why we Integrate joint density function with opposite bounds to get marginal density

I have a function $f_{x,y}(x,y)$ which represents the joint density function. In order to get marginal density function in terms of $x$, I need to integrate using $y$ bounds. Why is that? I assumed ...
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### 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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### 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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