Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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.
gaston
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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 want to loose meaning while doing this. Wikipedia…
David LeBauer
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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 marginal densities) upper bound for the copula…
Coppola
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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 have spent the last few months casually reading…
stats_noob
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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 < x < 1 and 0 < y < x}\\ 0 &…
soren.qvist
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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 for independence of $X$ and $Y$? I checked a…
Silverfish
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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, namely for $j=1,\ldots,K$: $$ S_j =…
R S
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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 product of the two marginal distributions. I have…
Rachel
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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 $p = f \cdot m$, for a density $f$ and Lebesgue measure…
Charac
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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 the geometrical solutions but what i would like to see…
math_law
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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} \sum_{i=1}^n…
Colin T Bowers
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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 the product formed by taking the product of the…
wnoise
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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 $(X_1,X_2)$ and marginal pdf of $X_1$. $$…
user1102
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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 $d$ degrees of freedom equal to the number of…
jmilloy
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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 $U_0=0$. I am trying to figure out the joint…
user13154
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