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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18 votes
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Why don't we see Copula Models as much as Regression Models?

Is there any reason that don't see Copula Models as much as we see Regression Models (e.g. https://en.wikipedia.org/wiki/Vine_copula, https://en.wikipedia.org/wiki/Copula_(probability_theory)) ? I ...
stats_noob's user avatar
16 votes
2 answers
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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 < ...
soren.qvist's user avatar
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 ...
David LeBauer's user avatar
16 votes
1 answer
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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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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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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 = ...
Charac's user avatar
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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 $...
jmilloy's user avatar
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3 answers
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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, ...
R S's user avatar
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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} \...
Colin T Bowers's user avatar
10 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 \...
jschnieder's user avatar
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 ...
math_law's user avatar
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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 ...
Rachel's user avatar
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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 ...
wnoise's user avatar
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2 answers
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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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9 votes
6 answers
10k views

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 ...
cgo's user avatar
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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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9 votes
4 answers
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Explanation of Joint Probability and Independence

Can anyone explain further to me the solution for the second instance where $f(x,y) = 24xy$. Specifically, I don't really understand the part "because the region in which the joint density is nonzero ...
user277416's user avatar
9 votes
1 answer
219 views

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: $$ F_{X,Y}(x,y)=G_1(x)G_2(y),\forall (x,y)\in\mathbb{R}\times\mathbb{R} $$ ...
Guilherme Salomé's user avatar
8 votes
3 answers
11k 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\{...
Baidal Kocham's user avatar
8 votes
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 ...
D1X's user avatar
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1 answer
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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 ...
Awani Khodkumbhe's user avatar
8 votes
1 answer
1k views

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 ...
Flying pig's user avatar
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8 votes
2 answers
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Derivative of the Joint Distribution Interpretation

Given two continuous random variables $X$ and $Y$, the joint cumulative distribution function $F_{X,Y}$ is defined as: $$F_{X,Y}(x,y)=\mathbb{P}(X\le x, Y\le y)=\displaystyle\int_{-\infty}^{x}\int_{-\...
Supreeth Narasimhaswamy's user avatar
8 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 ...
Eli's user avatar
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1 answer
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Expectation of the maximum of two correlated normal variables

I am curious what the derivation for the expectation of the maximum of two jointly normal random variables $X$ and $Y$ with correlation coefficient $\rho$. I could start with the following but the ...
ambushed's user avatar
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1 answer
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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}...
Adrian's user avatar
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8 votes
4 answers
336 views

The law of total probability: $P(T > t, Z = 1) = \int_t^\infty P(\cap_{j = 2}^k \{ T_j > x \} ) \lambda_1 e^{- \lambda_1 x} \ dx$?

I am studying Markov processes with exponential wait times. The following is said: Assume there are $k$ point events, denoted $w_1, \dots, w_k$, that the waiting time for $w_i$ to occur is $T_i \sim \...
The Pointer's user avatar
  • 1,902
8 votes
2 answers
208 views

Joint distribution of $Y$ and $S^2-Y^2$

Let $\{X_i\}_{i=1}^n\overset{iid}{\sim}\mathcal{N}(\mu,\sigma^2)$. Let $\{b_i\}_{i=1}^n$ be a sequence of numbers so that $\sum_{i=1}^nb_i=0$ and $\sum_{i=1}^nb^2_i=1$. Define $$S^2=\sum_{i=1}^n(X_i-\...
Tan's user avatar
  • 1,479
8 votes
1 answer
935 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 ...
Martin Modrák's user avatar
8 votes
1 answer
197 views

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....
Copuleros's user avatar
  • 337
8 votes
2 answers
998 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 ...
argan's user avatar
  • 81
7 votes
2 answers
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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 ...
notrockstar's user avatar
7 votes
2 answers
265 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 $$ f(x)=2x\ \ \ 0<x<1$$...
Qwerty's user avatar
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7 votes
2 answers
2k views

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)...
Remy's user avatar
  • 289
7 votes
2 answers
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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 ...
ColorStatistics's user avatar
7 votes
1 answer
1k 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 ...
user76284's user avatar
  • 983
7 votes
1 answer
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KL divergence for joint probability distributions?

I have a pair of joint probability distributions. I want to measure their similarity/dissimilarity. If they were single-dimensional probability distributions, then I could measure the Kullback–Leibler ...
rhombidodecahedron's user avatar
7 votes
2 answers
1k views

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 ...
Cornel's user avatar
  • 71
7 votes
2 answers
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How can I calculate a joint distribution based on marginal and conditional information?

Suppose I am given a probability mass function. A second one is also specified conditional on the first. Together, they must define a joint probability distribution. How can I calculate it?
Jxson99's user avatar
  • 664
7 votes
1 answer
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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:...
Toney Shields's user avatar
7 votes
1 answer
4k views

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_{...
Barranka's user avatar
  • 205
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 ...
LonelyBear's user avatar
7 votes
1 answer
6k views

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?
lpchristine's user avatar
7 votes
1 answer
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Predictive Posterior Distribution of Normal Distribution with Unknown Mean and Variance

Suppose that $x_{i}|\mu,\sigma^{2} \sim \mathcal{N}(\mu,\sigma^{2})$ for $i = 1,...n$. Assume that the assigned prior distributions are $\mu$ ~ $\mathcal{N}$($\mu_{0}$, $\sigma^{2}_{0}$) and $\tau \...
StatBeginner's user avatar
6 votes
2 answers
6k views

How to find conditional distributions from joint

I want to learn about how to do Gibbs sampling, starting with finding conditional distributions given a joint distribution. While looking for examples, I found this blog post that I wanted to ...
bill_e's user avatar
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6 votes
2 answers
2k 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 ...
Nancy's user avatar
  • 61
6 votes
1 answer
120 views

How to derive the joint distribution in these 3 models?

Carlos Cinelli in this great post https://stats.stackexchange.com/a/384460/198058 gives an example of 3 different Data Generating Processes/Causal Models giving rise to the same joint distribution $(X,...
ColorStatistics's user avatar
6 votes
1 answer
4k views

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}{\...
Kolibris's user avatar
  • 615
6 votes
2 answers
744 views

Exchangeability and joint distribution

The definition of an exchangeabilty for a finite sequence says that, if we have random variables $X_1,\ldots,X_n$, then for each permutation $\pi: \{1,\ldots,n\}\rightarrow\{1,\ldots,n\}$, the joint ...
Mentossinho's user avatar

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