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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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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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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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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14 votes
3 answers
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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 ...
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12 votes
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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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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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11 votes
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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2 answers
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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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11 votes
3 answers
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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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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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1 answer
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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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6 answers
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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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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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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
1 answer
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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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9 votes
1 answer
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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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9 votes
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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8 votes
3 answers
9k 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\{...
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8 votes
3 answers
15k views

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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8 votes
1 answer
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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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8 votes
4 answers
640 views

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 ...
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8 votes
1 answer
613 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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8 votes
4 answers
307 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 \...
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  • 1,126
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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8 votes
1 answer
191 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....
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8 votes
2 answers
817 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 ...
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7 votes
2 answers
8k views

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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7 votes
2 answers
251 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 $$ f(x)=2x\ \ \ 0<x<1$$...
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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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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)...
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  • 269
7 votes
2 answers
567 views

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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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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  • 913
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 ...
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7 votes
1 answer
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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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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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7 votes
1 answer
2k views

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 \...
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7 votes
0 answers
86 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-\...
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  • 1,349
6 votes
2 answers
4k 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 ...
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  • 2,701
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 ...
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6 votes
1 answer
3k 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}{\...
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  • 615
6 votes
2 answers
413 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 ...
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6 votes
2 answers
6k views

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_{-\...
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6 votes
2 answers
495 views

exact form for the marginal posterior

I have a question that I come across for practicing. Basically the question is this: Consider a random sample from the normal distribution with unknown mean and variance $Y_i \sim^{i.i.d.} N(\mu, \...
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  • 629
6 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:...
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6 votes
2 answers
2k views

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 ...
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  • 239
6 votes
3 answers
2k 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 ...
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  • 2,042
6 votes
1 answer
12k views

From joint cdf to joint pdf

We can get the joint pdf by differentiating the joint cdf, $\Pr(X\le x, Y\le y)$ with respect to x and y. However, sometimes it's easier to find $\Pr(X\ge x, Y\ge y)$. Notice that taking the ...
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  • 583
6 votes
1 answer
174 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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6 votes
1 answer
186 views

Showing independence between two functions of a set of random variables

I've been working on the following problem and I'm confused about how to get started: Let $X_1, X_2,\dots, X_n$ denote i.i.d. real valued random variables, each absolutely continuous with an ...
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