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Questions tagged [joint-distribution]

Joint probability distribution of several random variables gives the probability that all of them simultaneously lie in a particular region.

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Wavenet joint probability

As presented in the first article of Google Wavenet (https://arxiv.org/pdf/1609.03499.pdf) the model can approximate the joint probability of the whole sequence (raw audio waveform) using the chain ...
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$k$-th order statistics when the value of $j$-th one is known

Suppose there are $n$ random variables $X_i,~i\in\{1,\cdots,n\}$ which are independently drawn according to a CDF $F$ and pdf $f$. Suppose also that we know one of the realization, say $X_{(j)}=x_{(...
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joint PMF in a class of n students

A class of n students takes a test in which each student gets an A with probability p, a B with probability q, and a grade below B with probability 1 − p − q, independently of any other student. If X ...
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PMF and independence with two discrete random variables?

Each of n people (whom we label 1, 2, . . . , n) are randomly and independently assigned a number from the set {1, 2, 3, . . . , 365} according to the uniform distribution. We will call this number ...
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Probability of successful match with two faces in image

A photo containing faces of two different people is compared to labeled images of faces in database. The probability of a match on the first person is .70. The probability of a match on the second ...
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Joint Probability Distribution and covariance

If $$f(x,y)=1/4 $$ $x=-3,y=-5; x=-1,y=-1; x=1,y=1; x=3,y=5. $Find cov (x,y). I know the formula for cov (X,Y) but I'm stuck at finding E (x) and E (y).
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Marginal Distribution from Bivariate Distribution Matrix

I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The ...
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How to compute a 2D distribution function from its density?

Suppose that we have two random variables $X, Y$ with a joint probability density function $$f(x,y)=1,\ -y\lt x\lt y,\, 0\lt y\lt 1.$$ How can I calculate the cumulative joint probability function $...
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Having difficulty deciding limits of integration for a joint to marginal pdf

A joint pdf, $f_{X,Y}(x,y)=5$, is given with the following intervals: $-1<x<1$ $x^2<y<x^2+{1\over{10}}$ I am trying to find marginal pdf of $f_Y(y)$ but I am stuck. Trying for hours....
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Visualizing separability / independence

I’d like to visually ‘see’ the independence of random variables. I tried plotting f(x), f(y), and f(x, y) for independent and dependent pairs of variables. However, the difference is still not ...
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Joint cumulative distribution of independent random variables

X,Y,Z are non negative random variables which are independent and uniformly distributed in [0,1] and let $\alpha$ be a given number in [0.1]. Now how to compute $\text{Pr}(X+Y+Z>\alpha \;\;\; \&...
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Joint distribution of a two part model

Let $ Y $ be a random variable defined on $ (0, +\infty) $. In a univariate two part model, the distribution of $ Y $ is defined as follows \begin{equation*} g ( y_i ) = \left\{ \begin{array} { ...
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I am confused between selection of limits for joint distribution [duplicate]

I am given the following joint distribution of two random variables X and Y: $$ f(x,y)=2y \\ 0<y<1, 0<x<y $$ Now I need to find the marginal PDF of X but how do I ...
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What is the Joint Density Function of a Three-Level Mixed-Effects Model?

This is a follow-up question to a question I posted earlier. Obviously, maximum-likelihood estimation of mixed-effects models requires the joint density function. Let us assume a two-level mixed ...
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Multi-dimensional CDF on a discrete support

Suppose I have two discrete-support random variables, $X$ and $Y$. They have joint CDF $F(X,Y)$. If I want to find $\Pr(a \leq X \leq b , c \leq Y \leq d)$. It is obviously not: $F(b ,d)-F(a-1 ,...
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Deriving the joint probability density function from a given marginal density function and conditional density function

We have a multivariate normal vector $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, where $\mathbf{X} = \left[ \begin{matrix} X_1 \\ X_2 \\ \dot\\\\ \dot\\\\ X_n \end{matrix} \...
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GMM model of the joint distribution from multivariate marginals

I have two multivariate Gaussian variables $X \sim \mathcal{N}(\boldsymbol {\mu}_X \in \mathcal R^d,\boldsymbol {\Sigma}_X \in \mathcal R^{d \times d})$ and $Y \sim \mathcal{N}(\boldsymbol {\mu}_Y \...
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Identically distributed vs P(X > Y) = P(Y > X)

I've two related propositions which seem correct intuitively, but I struggle to prove them properly. Question 1 Prove or disprove: If $X$ and $Y$ are independent and have identical marginal ...
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How to specify joint distribution when they are not jointly independent?

I have three variables, $X, Y, Z$ with marginal distributions $F_x, F_y, F_z$. I want to specify the joint distribution $F_{xyz}$ of $(X, Y, Z)$. I know that if $X, Y, Z$ are jointly independent, ...
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What do the joint distributions look like? [duplicate]

I know that if I know the marginal distributions, that's not enough to specify the joint distribution. But obviously it can't be "any" joint distribution, it still needs to respect its marginal ...
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Maximum likelihood joint probability distribution (discrete & continuous)

I am trying to find the values $v_1$ and $v_2$ that maximizes the likelihood of some observations. I have information about $v_1$ and $v_2$ from a set of 'experiments'. In each experiment, $v_1$ and ...
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Resample from data with constraints to the marginal distribution

Motivation This problem comes from the situation where I have a non-random sample of individuals for which $p$ variables are measured. The target is to extract a subset of individuals which would be ...
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Confusion about range of integration for density function

Consider the joint density function: $$f(x,y) = \begin{cases} 2 & & \text{for } 0 \leq x \leq1 \text{ and } 0 \leq y \leq 1-x, \\[6pt] 0 & & \text{otherwise}. \end{cases}$$ From ...
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Is this sentence referring to joint or conditional probability?

The following is a quote from my textbook, in a chapter discussing the Viterbi algorithm (Durbin, Richard, et al. Biological sequence analysis: probabilistic models of proteins and nucleic acids. 1st ...
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When $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim(X_0-X_1, X_0-X_2)$?

Consider a bivariate probability distribution $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question: Are there necessary and sufficient conditions on the CDF associated with $P$ (joint or ...
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Shouldn't the joint probability of 2 independent events be equal to zero?

If the joint probability is the intersection of 2 events, then shouldn't the joint probability of 2 independent events be zero since they don't intersect at all? I'm confused.
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How to derive joint CDF Gumbel distribution

If you have 3 random variables: $X$, $Y$, and $Z$ and they have independent Gumbel distribution. $A$, $B$ and $C$ are three discrete random variables that are functions of $X$, $Y$, and $Z$ as per the ...
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Is “joint probability” assumption necessary for regression purposes?

I state beforehand that my question may sound odd and captious (and maybe it is). In regression theory basically we assume that explanatory variables and independent variable are joined togheter ...
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Finding joint support of $(XY,X/Y)$ where $(X,Y)$ has joint pdf $1/x^2y^2$ for $x,y\ge1$

$X$ and $Y$ are random variables with joint pdf $$\frac{1}{x^2y^2}\qquad,\, x\ge1,y\ge1$$ Set $$U=XY, V=X/Y$$ Explain why the joint range of $U$ and $V$ is given by: $$\{(u,v):v\in(0,1),u\ge1/v\} \...
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Calculating a probability based on a joint distribution between a Uniform random variable nested within a Uniform(0,1) random variable

Let $X_1 \sim Uniform(0,1)$, and $X_2 \sim Uniform(0, x_1)$, where $x_1$ is the realized value of $X_1$. Find $P(X_1 + X_2 \geq 1)$. I know that I need the joint distribution of $X_1$ and $X_2$. $...
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1answer
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calculation of paramters needed for joint probability distribution?

Please correct me if I'm wrong. From my understanding, the number of entries in the above image is 7 because you need to calculate 7 and the 8th one can be done by 1-p. But I can't understand how ...
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To find the covariance given the joint probability density function.

Question: I was solving some question papers and got stuck in this problem. My problem: I know how to find "marginal probabilities" from a joint probability density function and also know how to ...
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Is there an analytical way of determining the probability for a specific outcome given a joint p.m.f.?

I'm looking for a way to determine the probability for a specific outcome based on (what I think should probably be) a joint probability mass function. I'll try to put into words my specific case: I ...
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1answer
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Should I use another random variable for the sum, if I work with random vectors?

Assume I have a simple random vector $(X, Y)$ with common distribution $P((0, 0))=1/6, P((1, 1))=1/6, P((3, 1))=1/4, P((0, 2))=1/6, P((1, 2))=1/4$, all others are zero. If I would like to argue ...
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how to find correlation coefficient when X and Y follows Poisson Distribution?

A bridge is examined for corrosion. It is believed that the corrosion on left side exist is poisson distributed with mean 3 and corrosion on right side is poisson distributed with mean 1.5+0.5X where ...
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Uncorrelatedness + Joint Normality = Independence. Why? Intuition and mechanics

Two variables that are uncorrelated are not necessarily independent, as is simply exemplified by the fact that $X$ and $X^2$ are uncorrelated but not independent. However, two variables that are ...
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Loss function for comparing high-dimensional joint distributions

I'm synthesizing data trained from a source dataset, and am looking for a loss function to compare different data synthesis methods*. I have some ideas below, but each has drawbacks and none is very ...
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Can the marginal distributions of A,C and B,C be used to build joint distribution of A and B?

There are three random variables $A$, $B$ and $C$. If the variables $A$ and $B$ were independent, their marginal joint distribution would be given by $$ P(A,B) = P(A)P(B) $$ For example, given the ...
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How is exchangeability related to covariate shift?

I understand that exchangeability refers to the notion that the order of data in a sequence does not affect the joint distribution of that data. In a sense, the current data we possess is from the ...
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How does one write joint p.d.f., when the parameter (e.g. $\theta$) is a vector $\theta=(\theta_1,…,\theta_n)$?

How does one write joint p.d.f., when the parameter (e.g. $\theta$) is a vector $\theta=(\theta_1,...,\theta_n)$? And each $\theta_i$ is meant to correspond like $X_i \sim SomeDistr(\theta_i)$ Assume ...
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Limits of integration of a density function

My question is based on this post. In summary, $X \sim \text{Unif}(a,b)$ and $Y|X \sim \text{Unif}(a,X)$. Then the author does the following calculations: \begin{align} f(y) = \int_{-\infty}^{\infty} ...
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Conditional Expectation of pdf

Wish to identify what I'm doing wrong when finding the $\operatorname E(X\mid Y=5)$ of the following: $$f(x, y)=\begin{cases} 1/6 & \text{if } 0<x<2, 0<y<6-3x \\ 0 & \text{...
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1answer
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Expected vs observed frequency of two events at the same time

I'll first give an example and afterwards a more formal definition of my problem. Example: Let's assume I'm looking at balls with two properties: their color can be black or white, their weight ...
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1answer
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Showing indepedence of two random variables when $p(x,y) = p(x) \cdot p(y)$ except a constant factor?

During a course I attend at university, I encountered the following question: Given is a probability distribution: $$p(x,y) = \lambda \eta \cdot \exp(-\lambda x - \eta y) $$ supported on $\mathbb{R}...
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2answers
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Variance being negative

Let $X$ and $Y$ have joint pdf such that $$f(x,y) = 3e^{-3x-y}, 0 < x< \infty, 0< y< \infty.$$ (a) Show that $X$ and $Y$ are independent. (b) Calculuate $Var(X)$. In ...
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Joint probability distribution of geometric distribution

Let $X$ and $Y$ be independent and identically distributed $(i.i.d.)$ r.v.’s, each having the probability distribution, $p(k) = (1 − λ)λ^k$; $k = 0,1,...$ where $λ :(0; 1)$ is a constant. Define $U = ...
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When are correlated Normal random variables multivariate Normal?

I know that there are many example of correlated normal random variables which are not jointly (multivariate) normal. However, are there conditions which state when correlated normal random variables ...
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260 views

Joint probability of multivariate normal distributions with missing dimensions

Suppose I conduct two experiments, each measuring a subset of possible parameters. From experiment #1 I measure two parameters and estimate the multivariate normal distribution $$ \mathbf{X}_1=\left [...
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1answer
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Applying assumptions about marginal and conditional PDFs

We are given $0 < x_2 < x_1 < 1$. What assumptions can you make about $f_1(x_1)$ and $f_{2|1}(x_2|x_1)$? I know that $f(x_1) f_{2|1}(x_2|x_1) = \frac{1}{x_1}$. I know the expression can be ...
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Is it possible to find the joint distribution of a random vector if only the distribution of scalar many-to-one transformation is known? [duplicate]

Theoretical Exercise: I'd like to derive the joint distribution $p_{\boldsymbol{X}}$ of a random vector $\boldsymbol{X} \in \mathbb{R}^K$ if only the distribution of a scalar many-to-one ...