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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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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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Upper bounds for the copula density?

The Fréchet–Hoeffding upper bound applies to the copula distribution function and it is given by $$C(u_1,...,u_d)\leq \min\{u_1,..,u_d\}.$$ Is there a similar (in the sense that it depends on the ...
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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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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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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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Does the multivariate Central Limit Theorem (CLT) hold when variables exhibit perfect contemporaneous dependence?

The title sums up my question, but for clarity consider the following simple example. Let $X_i \overset{iid}{\backsim} \mathcal{N}(0, 1)$, $i = 1, ..., n$. Define: \begin{equation} S_n = \frac{1}{n} \...
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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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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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1answer
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Why autocovariances could fully characterise a time series?

I read in John Cochrane's Time Series for Macroeconomics and Finance that: Autocovariance can fully charaterize the time series [joint distribution]. I do not fully understand the connection ...
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1answer
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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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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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X,Y univariate random variable with $F_{X,Y}(x,y)=G_1(x)G_2(y)$: are they independent?

Let $X:\Omega\to\mathbb{R}$ and $Y:\Omega\to\mathbb{R}$ be univariate random variables with CDF $F_{X,Y}(x,y)$ such that: $$ F_{X,Y}(x,y)=G_1(x)G_2(y),\forall (x,y)\in\mathbb{R}\times\mathbb{R} $$ ...
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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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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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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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1answer
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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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intuitive difference between joint probability and conditional probability in this example

I was reading a tutorial on marginal densities when I came across this example (rephrased). A person is crossing the street and we want to compute the probability when he gets hit by a passing car ...
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2answers
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Joint distribution of two gamma random variables

I am so puzzled by this problem. Given two variables $X_1$ and $X_2$, such that $X_i \sim \mathrm{Gam}(a_i, b)$, find the joint distribution of $X_1$ and $X_2$. I understand how to proceed if ...
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McKay's bivariate Gamma distribution

Given the variables $X$ and $Y$, which are correlated, $X\ge0$, $Y\ge0$ and each follow a gamma distribution with different shape parameters, i.e.,$X\sim\Gamma(a_1,\alpha)$ and $Y\sim\Gamma(a_2,\alpha)...
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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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Question about joint distribution of Bernoulli random variables under constraint that sum must be 1

I am stuck with a problem at work. Can anybody please help me to give me the joint distribution of $n$ Bernoulli random variables but under the constraint that the sum of the these $n$ random ...
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3answers
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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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1answer
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Expected value of product of non independent Bernoulli random variables (correlations are known)

I've asked a question about getting the joint probability distribution for $N$ Bernoulli random variables, given the expected value for each one ($E[X_i]=p_i)$ and it's correlations ($\rho_{12},\rho_{...
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1answer
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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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1answer
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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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1answer
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Why is this representing the left tail?

In this source about the Clayton copula on page 18 they write: It has been used to study correlated risks because it exhibits strong left tail dependence and relatively weak right tail dependence....
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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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1answer
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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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3answers
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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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1answer
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From Marginal Exp-Norm Distributions to What Conditionals and Joint?

I have (trivariate: multivariate with three variables) data that appears to be good empirical and reasonable theoretical fit for a (univariate) convolution of an exponential and a normal distribution (...
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1answer
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Degenerate random variable

Let $X$ and $Y$ be independent $rv$ such that $XY$ is a degenerate $rv$. Can I say that individually $X$ and $Y$ are also degenerate? Why?
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3answers
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Joint pdf of a continuous and a discrete rv

Let us consider a manufacturing system. It involves 2 independent components. If one of these components fails then the entire system fails. Let $Y_j$ be distributed $\exp(Q_j)$ where $j=1, 2$. If ...
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2answers
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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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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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1answer
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Bivariate normal distribution and its distribution function as correlation coefficient $\rightarrow \pm 1$

I am not sure what happens to a bivariate normal distribution when $|\rho| \rightarrow 1$. Is the distribution well defined then? Moreover, when $$ \Phi \left(\frac{x_1}{\sigma_1}, \frac{x_2}{\...
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1answer
803 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 ...
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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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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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1answer
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Does permutation permute also dependence?

I have a random vector $X = (X_1, \ldots, X_n)$ jointly distributed in someway, assuming also some mutual dependence between its marginal variables. If I apply a permutation to the vectors drawn from ...
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4answers
269 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
70 views

Are two Random Variables Independent if their support has a dependency?

This might be a really dumb question, but in a joint PDF of $X$ and $Y$, $f_{XY}(x,y)$, if the support of a random variable $Y$ depends on $X$, are the two random variables necessarily dependent? For ...
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1answer
416 views

How to test the exchangeability of data?

I have implemented a constraint-based random number generator producing 3 columns with the constraint that each row sums to 1: ...
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1answer
157 views

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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1answer
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Sums of Random Variables from Order Statistics of Dice Rolls

Let's say you have a set of order statistics $ X_{(1)}, \dots, X_{(N)} $ drawn from a discrete uniform distribution $ \text{unif}(1,S) $. If you choose $ X_{(n_1)}, \dots, X_{(n_k)} $ from this set, ...
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2answers
269 views

Expected value of $Ye^X$ where $X \sim U(0,1)$ and $Y \sim U(0,1)$

I am trying to find the expected value of $Z$ where $Z = Y\cdot e^X$ where $Y \sim U(0,1)$ and $X \sim U(0,1)$. My attempt so far: $$F_Z(z) = P(Ye^X \le z) = \int \int_{Ye^X \le z} f(x,y)\, dxdy$$ ...
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1answer
127 views

Why might it be important to specify a family when using GEE if it doesn't make any assumption about the joint distribution?

I understand that one of the advantages of GEE is that you make no assumptions about the joint probability distribution, relying instead on the mean, the variance and the associations (corr). Why then ...
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1answer
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Expected value of $\dfrac 1 {I(y_1<c) + I(y_2<c)}$, where $y_1$ and $y_2$ are i.i.d. random variables with exponential distribution

There is a random variable $r = \dfrac 1 {I(y_1<c) + I(y_2<c)}$. Both $y_1$ and $y_2$ are i.i.d. random variables with exponential distribution, so their joint distribution is $f(y_1, y_2) = \...
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1answer
186 views

Distribution of sufficient statistics for right-censored exponential data

Let's assume we only have right-censoring after a fixed time $\tau$, then for i.i.d. exponentially distributed failure times $X_i \sim \text{Exp}(\lambda)$ the sufficient statistics (under the clearly ...
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1answer
159 views

How to calculate $P (|X − Y | ≤ 1/6)$? [duplicate]

$f_{X,Y} \left( x, y \right) = 1\quad \text{for}\quad 0≤x≤1,\ 0≤y≤1 $ and $0$ otherwise. How to calculate $P \left( |X − Y | ≤ 1/6 \right)$?
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1answer
674 views

Joint distribution of two sums of correlated variables

Suppose that $(X_1, Y_1)$ and $(X_2, Y_2)$ are independent and have the same joint distribution $F_{X,Y}$, which is a known copula $C_{X,Y}(F(X), F(Y))$. Also, suppose that $V = X_1 + X_2$ and $W = ...