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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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 ...
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Distribution given sum

I'm stuck on an exercise (it's not homework, but preparation for finals). It goes like this: $X_1, \dots, X_n$ are iid Exponential($\lambda$) (with parametrization $f(x)=\lambda e^{-\lambda x}$). What ...
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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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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 ...
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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 ...
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5 votes
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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 ...
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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 ...
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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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2 answers
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Difference between joint density and density function of sum of two independent uniform random variables

I am not able to understand the difference between the joint density function and density function for a random variable $Z = X_1 + X_2$, where $X_1, X_2$ are uniform random variables in $[0,1]$. I ...
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2 answers
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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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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 ...
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9 votes
1 answer
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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=...
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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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2 answers
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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 ...
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2 answers
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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 $...
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3 answers
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What is the number of parameters needed for a joint probability distribution?

Let's suppose we have $4$ discrete random variables, say $X_1, X_2, X_3, X_4$, with $3,2,2$ and $3$ states, respectively. Then the joint probability distribution would require $3 \cdot2 \cdot2 \cdot ...
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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\{...
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6 votes
3 answers
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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 ...
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9 votes
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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 $(...
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7 votes
1 answer
680 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 ...
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1 vote
2 answers
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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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7 votes
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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 $$ f(x)=2x\ \ \ 0<x<1$$...
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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 ...
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  • 153
4 votes
2 answers
636 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 ...
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3 votes
2 answers
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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 ...
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2 votes
1 answer
467 views

Proof Verification: Joint variance of the product of a random matrix with a random vector

BACKGROUND QUESTIONS Is the proof of my claim correct? How might my proof be improved? Claim: (1) The joint-covariance matrix of the product of a real random matrix $X$ of dimension $v\times m$...
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1 vote
1 answer
115 views

What is the expectation of the following joint CDF?

Yesterday, I asked the following question regarding copulas: "Let's say $X=(X_1,X_2)′$, where $X\in \mathbb R^2$. What is the expectation of the copula function $C(F_{X_1}(x_1),F_{X_2}(x_2))$ - i.e....
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Convergence in Distribution in Order Statistics

Let $X_1, X_2, \ldots$ be iid from Exp$(\theta)$ with density function $f(x) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$. (a) Find the limiting distribution of $M_n = Y_1 - \theta\ln(n)$ and $T_n = nY_n$,...
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33 votes
3 answers
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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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9 votes
3 answers
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Proof that joint probability density of independent random variables is equal to the product of marginal densities

Is it true that if $X_1, X_2, \ldots ,X_n$ are independent random variables, then \begin{align} & f_{X_1,X_2,\ldots,X_n}(x_1,x_2,\ldots,x_n) \\ = {} & f_{X_1}(x_1)\times f_{X_2}(x_2) \times \...
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15 votes
2 answers
64k 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 < ...
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16 votes
5 answers
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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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7 votes
1 answer
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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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9 votes
1 answer
4k 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 $...
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5 votes
3 answers
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What is the joint probability distribution of two same variables

Let $f\left(x\right)$ be the probability density function of the random variable $X$. What is the joint probability distribution of $f_{X,Y}\left(x,y\right)$ if $Y=X$? Thanks for any helpful answer.
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2 answers
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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 ...
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5 votes
1 answer
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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 $X_i$? ...
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4 votes
2 answers
995 views

Interpretation of cartesian product of the support of marginal distribution

Suppose we have a multivariate data set, $s = (s_1, s_2, ... s_p)$ and each $s_i$ is distributed with a distribution that has finite support (we'll call each $s_i$ a "source"). Let us ...
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3 votes
1 answer
338 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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  • 1,055
2 votes
1 answer
3k views

P(A|B) and P(A|C) known, what is P(A|BC)

Using: $$ \begin{align} P(A =1|B=1) &= 0.9\\ P(A =1|C=1) &= 0.9\\ P(A=1) &= 0.5 \end{align} $$ I want to know the probability of $P(A=1|B=1,C=1)$ I also know that $B$ and $C$ and ...
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  • 23
11 votes
3 answers
7k 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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  • 581
10 votes
1 answer
1k views

Is the maximum entropy distribution consistent with given marginal distributions the product distribution of the marginals?

There are generally many joint distributions $P(X_1 = x_1, X_2 = x_2, ..., X_n = x_n)$ consistent with a known set marginal distributions $f_i(x_i) = P(X_i = x_i)$. Of these joint distributions, is ...
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  • 515
8 votes
1 answer
594 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 ...
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7 votes
1 answer
1k views

Joint distribution in layman's terms

Can someone please explain to me in layman's terms what a joint distribution is? I do not understand it after seeing a word problem that pertained to joint distributions. Please provide the intuition ...
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2 votes
1 answer
181 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 $f_{X,Y}$, ...
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  • 2,133
2 votes
1 answer
775 views

Joint probability measure

I know from my measure theory class that for two $\sigma$-finite measure spaces $(\mathcal{X}_1, \mathcal{A}_1, \mu_1)$ and $(\mathcal{X}_2, \mathcal{A}_2, \mu_2)$ there exists a unique measure $\mu :=...
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  • 597
12 votes
1 answer
6k 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 ...
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10 votes
4 answers
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Product of 2 Uniform random variables is greater than a constant with convolution

I am trying to formulate the following question. X and Y are IID , uniform r.v. with ~U(0,1) What is the probability of P( XY > 0.5) = ? 0.5 is a constant here and can be different. I do respect ...
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  • 231
6 votes
1 answer
1k 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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4 votes
0 answers
831 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 ...
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