# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 $Pr\{... 1answer 3k views ### Bivariate normal distribution and its distribution function as correlation coefficient$\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 $$\Phi \left(\frac{x_1}{\sigma_1}, \frac{x_2}{\... 2answers 4k views ### What is the number of parameters needed for a joint probability distribution? Let's suppose we have 4 discrete random variables, say X_1, X_2, X_3, X_4, with 3,2,2 and 3 states, respectively. Then the joint probability distribution would require 3 \cdot2 \cdot2 \cdot ... 1answer 816 views ### How can one construct a cumulative probability distribution function from 2 others? I dip into project time estimation and can't find intuition. What is the cumulative probability distribution of an event when two independent tasks both complete successfully (when performed in ... 2answers 184 views ### Identically distributed vs P(X > Y) = P(Y > X) I've two related propositions which seem correct intuitively, but I struggle to prove them properly. Question 1 Prove or disprove: If X and Y are independent and have identical marginal ... 1answer 82 views ### Does permutation permute also dependence? I have a random vector X = (X_1, \ldots, X_n) jointly distributed in someway, assuming also some mutual dependence between its marginal variables. If I apply a permutation to the vectors drawn from ... 4answers 274 views ### Joint probability of multivariate normal distributions with missing dimensions Suppose I conduct two experiments, each measuring a subset of possible parameters. From experiment #1 I measure two parameters and estimate the multivariate normal distribution$$ \mathbf{X}_1=\left [... 1answer 71 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 ... 1answer 690 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}) $have a closed form? Edit:... 1answer 426 views ### How to test the exchangeability of data? I have implemented a constraint-based random number generator producing 3 columns with the constraint that each row sums to 1: ... 1answer 158 views ### Limits of integration of a density function My question is based on this post. In summary,$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} ... 1answer 200 views ### Integrating out a gamma-distributed parameter from a Weibull distribution I'm dealing with a variation of the three-parameter Weibull distribution where the third parameter is randomly distributed over a Gamma distribution. The cdf takes the form: $$G(x|\gamma) = 1-\exp\... 1answer 255 views ### Sums of Random Variables from Order Statistics of Dice Rolls Let's say you have a set of order statistics X_{(1)}, \dots, X_{(N)} drawn from a discrete uniform distribution \text{unif}(1,S) . If you choose X_{(n_1)}, \dots, X_{(n_k)} from this set, ... 2answers 271 views ### Expected value of Ye^X where X \sim U(0,1) and Y \sim U(0,1) I am trying to find the expected value of Z where Z = Y\cdot e^X where Y \sim U(0,1) and X \sim U(0,1). My attempt so far:$$F_Z(z) = P(Ye^X \le z) = \int \int_{Ye^X \le z} f(x,y)\, dxdy$... 1answer 128 views ### Why might it be important to specify a family when using GEE if it doesn't make any assumption about the joint distribution? I understand that one of the advantages of GEE is that you make no assumptions about the joint probability distribution, relying instead on the mean, the variance and the associations (corr). Why then ... 1answer 55 views ### Expected value of$\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) = \...
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 ...