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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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3answers
376 views

Conditional expectation of two identical marginal normal random variables

Let $Y_0$ and $Y_1$ be both identically (not necessarily independent) normally distributed with mean $\mu$ and $\sigma^2$, i.e., $Y_i \sim N(\mu, \sigma^2)$ for $i = 1, 2$. Let $\rho$ denote the ...
3
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1answer
3k views

Same Joint Distribution, different conditional and marginal distribution

I have a group of samples drawn from density function $p(x,y)$, so it has the marginal density $p(x)$ and $p(y)$, and conditional density $p(x|y)$ and $p(y|x)$. In what way I can construct another ...
6
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2answers
1k views

If any two variables follow a normal bivariate distribution does it also have a multivariate normal distribution?

Bivariate and multivariate distribution relationship. If we have say 3 variables where any two variables follow a normal bivariate distribution, then does it necessarily follow a multivariate ...
4
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1answer
604 views

Obtain marginal CDF from joint CDF by simulation

How can I evaluate the marginal cumulative distribution function of a set of random variables for which I do not have the CDF in closed form. I can, however, simulate from a joint distribution ...
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2answers
955 views

What is the moment of a joint random variable?

Simple question, yet surprisingly difficult to find an answer online. I know that for a RV $X$, we define the kth moment as $$\int X^k \ d P = \int x^k f(x) \ dx$$ where the equality follows if $p = ...
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1answer
4k views

What does it mean to factor a joint distribution?

A book I'm reading (Hogan & Mason, 2012, p37) contains the following passage: The joint distribution can be factored in two different ways into conditional and marginal probabilities that ...
8
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1answer
884 views

Spacings between discrete uniform random variables

Let $U_1, \ldots, U_n$ be $n$ i.i.d discrete uniform random variables on (0,1) and their order statistics be $U_{(1)}, \ldots, U_{(n)}$. Define $D_i=U_{(i)}-U_{(i-1)}$ for $i=1, \ldots, n$ with $U_0=...
7
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1answer
305 views

Is there a parametric joint distribution such that $X$ and $Y$ are both uniform and $\mathbb{E}[Y \;|\; X]$ is linear?

Is there a parametric joint distribution such that $X$ and $Y$ are both uniform on $[0, 1]$ (i.e. a copula) and $\mathbb{E}[Y | X = x]$ is linear (by which I mean affine) in $x$? That is, $$\mathbb{E}...
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2answers
889 views

Joint distribution of two dependent variables

I was wondering how to (or if it is even possible) find the continuous joint distribution between two random variables $x$ and $y$ when you know the continuous marginal density distributions of both $...
6
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3answers
186 views

When are correlated Normal random variables multivariate Normal? [duplicate]

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 ...
5
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3answers
3k views

Minimum CDF of random variables

I know that the joint cumulative function of two random variables X and Y is defined as: $F_{X,Y}(x,y)=P(X≤x,Y≤y)$. How can I find the CDF for $F_{X,Y}=\{x,x\}$. In other words is what will be $Pr\{...
4
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1answer
307 views

Maximum of a probability vector distributed as a Dirichlet variate

Let $p_1, p_2, \ldots \sim \text{Dirichlet}(\alpha_1, \alpha_2, \ldots)$. What is the distribution of $\max(p_1, p_2, \ldots)$? I have searched for the order statistics of the Dirichlet distribution ...
3
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2answers
100 views

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 ...
2
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2answers
409 views

Gibbs sampling and mixed distribution

For a project, I need to simulate from a joint distribution with both continuous and discrete variables that are dependent. The conditional distribution of any variable given the rest is known. I ...
30
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3answers
5k views

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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1answer
4k views

Estimating joint distributions using copula package in R

I am trying to estimate the joint distribution of stock returns using the copula package. I have read a couple of papers on copulae, but alas my lack of math ...
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5answers
4k views

Difference between the terms 'joint distribution' and 'multivariate distribution'?

I am writing about using a 'joint probability distribution' for an audience that would be more likely to understand 'multivariate distribution' so I am considering using the later. However, I do not ...
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1answer
30k views

How to find marginal distribution from joint distribution with multi-variable dependence?

One of the problems in my textbook is posed as follows. A two-dimensional stochastic continuous vector has the following density function: $$ f_{X,Y}(x,y)= \begin{cases} 15xy^2 & \text{if 0 < ...
8
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1answer
2k views

Mahalanobis distance on non-normal data

Mahalanobis distance, when used for classification purposes, typically assumes a multivariate normal distribution, and the distances from the centroid should then follow a $\chi^2$ distribution (with $...
3
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1answer
3k views

Marginal, joint, and conditional distributions of a multivariate normal

Let $Y$ ~ $MVN_3(\mu, \Sigma)$ where $\mu = (5,6,7)$ and $\Sigma = \begin{bmatrix}2 & 0 & 1\\0 & 3 & 2\\1&2&4\end{bmatrix}$ Find (a) The marginal distribution of $Y_1$ (b) The ...
7
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1answer
231 views

Analytically solving sampling with or without replacement after Poisson/Negative binomial

Short version I am trying to analytically solve/approximate the composite likelihood that results from independent Poisson draws and further sampling with or without replacement (I don't really care ...
6
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2answers
497 views

Why do copulas need the i.i.d assumption for marginal distribution?

Does anyone know if are there some assumptions for Copula method? I heard from someone that the data should be i.i.d (independent and identically distributed). Let's say, if I want to capture the ...
4
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1answer
2k views

How can I sample from a correlated multivariate Bernoulli distribution with known covariances?

I have $N$ Bernoulli random variables $X_1, ..., X_{N}$ with known parameters $p_1, ..., p_{N}$. They are dependent with known covariances. How can I sample from the joint distribution of the $X_i$? ...
3
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1answer
192 views

joint probability distribution of $k \le n$ order statistics

For $X_i \sim$ iid random variables: For $1\le r_1 < ..<r_k \le n$ integers, I am trying to find the joint pdf of: $$ (X_{(r_1)},...,X_{(r_n)}) $$ where $X_{(r_1)}$ is the $r_1$th largest ...
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1answer
2k views

Partial Derivative of Joint Distribution Function interpretation

Suppose we have \begin{equation} F(x,y) = \int_{-\infty}^x \int_{-\infty}^y f(a,b) \ db \ da \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1] \end{equation} From this, we can say the following: \begin{align} \...
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1answer
158 views

Derivation of conditional distribution from other two distributions

$$Y|X=x \sim N(x,1)\\X\sim N(\mu,\sigma^2 )$$ What distribution does $X|Y=y$ follow? My initial startegy was to $f_{Y|X}f_X=f_{X,Y}$ and solve for $f_{X|Y}=f_{X,Y}/f_{Y}$ . Computing for $f_{X,Y}$, ...
10
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1answer
3k views

Necessary and sufficient condition on joint MGF for independence

Suppose I have a joint moment generating function $M_{X,Y}(s,t)$ for a joint distribution with CDF $F_{X,Y}(x,y)$. Is $M_{X,Y}(s,t)=M_{X,Y}(s,0)⋅M_{X,Y}(0,t)$ both a necessary and sufficient condition ...
8
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2answers
212 views

Independence and Order Statistics

I have a problem at hand, which I am not being able to proceed. Can someone help me begin? $Y_1<Y_2<Y_3$ :An order statistic of size 3 from distribution having pdf $$ f(x)=2x\ \ \ 0<x<1$$...
7
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3answers
5k views

How to compare joint distribution to product of marginal distributions?

I have two finite-sampled signals, $x_1$ and $x_2$, and I want to check for statistical independence. I know that for two statistically independent signals, their joint probability distribution is a ...
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0answers
524 views

directed bayesian network and factor graphs

I have a directed bayesian given by the figure below. In the figure the circles are random variables and the shaded ones are observed. The rectangular nodes are constants representing the hyper ...
4
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1answer
666 views

Joint distribution of dependent Binomial random variables

Suppose we have $X_{1} \sim B(m,p_{1}), X_{2} \sim B(m,p_{2}),\cdots, X_{n} \sim B(m,p_{n})$ and they are dependent. Does the joint distribution $f(X_{1},X_{2},\cdots,X_{n}) $ have a closed form? Edit:...
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0answers
856 views

Non-parametric estimate of conditional expectation

I have a (fairly smooth) function $f$ and a sample $\{(x_i,y_i)\}_{i=1,\ldots,N}$ from the joint distribution of the random variables $X$ and $Y$. I would like to estimate the conditional expectation ...
3
votes
1answer
565 views

Copula for non-standard distributions in R

I'm trying to model a bivariate distribution using copulas in R. See image below for the pairs plot of the data. Variable 1 can be modeled nicely using a gamma distribution, but variable 2 fails ...
2
votes
1answer
153 views

Why can Gibbs sampling outputs be used in Rao-Blackwellization?

I'm currently learning Chib (1995)'s method to calculate the marginal likelihood of a Bayesian model using Gibbs sampling outputs. I'm stuck in the Rao-Blackwellization step. Suppose $\mu$ and $\phi$...
12
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3answers
2k views

Maximum likelihood estimator of joint distribution given only marginal counts

Let $p_{x,y}$ be a joint distribution of two categorical variables $X,Y$, with $x,y\in\{1,\ldots,K\}$. Say $n$ samples were drawn from this distribution, but we are only given the marginal counts, ...
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2answers
6k views

Problem calculating joint and marginal distribution of two uniform distributions

Suppose we have random variable $X_1$ distributed as $U[0,1]$ and $X_2$ distributed as $U[0,X_1]$, where $U[a,b]$ means uniform distribution in interval $[a,b]$. I was able to compute joint pdf of $(...
5
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1answer
806 views

How can one construct a cumulative probability distribution function from 2 others?

I dip into project time estimation and can't find intuition. What is the cumulative probability distribution of an event when two independent tasks both complete successfully (when performed in ...
4
votes
1answer
49 views

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 ,...
4
votes
1answer
445 views

How to show this curious combination of Exponential order statistics has a Chi-squared distribution?

Let $X_1, \ldots, X_n$ be i.i.d. exponentially distributed random variables with density $$\eqalign{\theta^{-1} e^{-x/\theta}, &x \ge 0 \\ 0, &x \lt 0} $$ and let $Y_i = X_{(i)}$ denote the ...
3
votes
2answers
557 views

Probability density of sum of two beta random variables

Suppose that $X$ and $Y$ are independent and have beta distributions. $X$ has probability density function $g(x)=6x(1-x)$ for $ 0\leq x \leq 1$ and $Y$ has the probability density function $h(y)=12y^2(...
3
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0answers
2k views

Generating samples from Copula in R

Suppose I want to model dependence between $d$ r.v.´s $Y_1,...,Y_d$ with the copula $C_\theta$, where $\theta$ are the corresponding parameters of that copula. I've also determined the correlation ...
3
votes
1answer
491 views

Relations between probabilities of “almost” independent random variables

Let $X$ and $Y$ be two random variables, such that the (average) mutual information is very small: $$ 0 \le I(X;Y) \le \epsilon \ll 1$$ In this case, we say that $X$ and $Y$ are almost independent. ...
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0answers
228 views

Distribution of ratio of two independent normal variables

My objective is to find out the distribution of $A/B$ given $A \sim N(a,b); B \sim N(c,d)$. I set $Z_1$ equal to $\frac{(A-a)}{\sqrt{b}}$ and $Z_2$ equals $\frac{(B-c)}{\sqrt{d}}$ such that $Z_1$ ...
2
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2answers
128 views

Is it possible to derive joint probabilities from marginals with assumptions about the conditionals?

I understand the title is too generic. I tried to look for similar questions and although there were a few that were seemingly about the same issue, either they provided answers in the negative or had ...
1
vote
1answer
2k views

trivariate normal distribution and joint distribution

Suppose that r.v $X$ and $Y$ are normally distributed and independent each other. Under which conditions $(X,Y)$ are bivariate normally distributed? Now I want to calculate $\text{Prob}(X>a, Y<a,...
0
votes
1answer
39 views

How to find CDF of a function of continuous joint distribution from PDF of joint distribution?

Here's what I think I should proceed: Make a level curve for the function keeping the constraints given in the PDF of joint distribution in mind. Find the area of interest keeping the constraints ...
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0answers
19 views

Level curves and functions of pair of Random Variables?

I came across the following question: I tried solving it, the following is my 1st attempt:(2nd method at the end) 𝑃[𝑊≤𝑤]=𝑃[𝑋𝑌≤𝑤]=𝑃[𝑌≤𝑤/𝑋] And then I simply double integrated keeping the ...
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0answers
130 views

Simulating correlated lifetimes at pre-specified correlation level

I am trying to simulate remission times for 100 patients from an exponential distribution with mean 1 year. I also want to simulate after-remission times for these 100 patients. But it is very much ...
0
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1answer
297 views

What is the conditional probability of variables in a multivariate gaussian?

Given $Y = [Y_1, Y_2, \ldots, Y_n]^T \sim N(0, \Sigma)$. That is $f_{Y}(y_1, y_2, \ldots, y_n)$ is a multivariate gaussian with mean $0$ and covariance matrix $\Sigma$. I'm asked to compute the ...