# 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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### Given pairwise marginal distributions, what can we say about the full joint?

From reading previous posts, I understand that, if we have pairwise marginals, say $P(A,B)$, $P(B,C)$, and $P(C,A)$, this doesn't allow us to reconstruct the full joint $P(A,B,C)$. But does it allow ...
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### Constructing Joint Multivariate Normal Distribution from Marginal Distribution (detrended data)

I have two n-length data vectors $X_1 \sim N(\mu_{1},\Sigma_{1})$ and $X_{2} \sim N(\mu_{2}, \Sigma_{2})$ which may or may not have a covariance. To see whether they do or not, I detrend them by ...
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### Confidence interval for a normal RV that its std is the mean of a different normal RV

Question: Suppose $X\sim\mathcal{N}(\mu_X,a),Y\sim\mathcal{N}(\mu_Y,b\cdot\mu_X)$ where $a$ and $b$ are known, but $\mu_X$ and $\mu_Y$ are not. What confidence bounds can we give for $\mu_Y$ from one ...
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### Joint probability based on assuming linear relationships

I am a statistical novice trying to work something out which is probably basic. Or impossible. I have discrete probability distributions for two random discrete variables $A$ and $B$. I want to find ...
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### Dependence or independence of three random variables

Consider I have three random variables A, B, C. I know that A depends on (B,C). Can I always deduce that it implies that A depends on B and also A depends on C? I mean does it implies that neither A ...
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### how to calculate conditional probablity when one even'ts occurance is dependent on mutiple events

I have X, Y, and Z all as binary variables, values either 0 or 1. Y and Z are and got values of P(Y = 1), P(Z = 1), P(X = 1|Y = 1, Z = 1) , P(X = 1|Y = 1, Z = 0) and P(X = 1|Y = 0). here I need to ...
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### Existence of Distribution with Given Multivariate Marginals

Consider discrete random variables $X_1,\cdots, X_n$, and let $D$ be their joint distribution. For each subset $S\subseteq[n]$ let $D_S$ be the marginal distribution $(X_i)_{i\in S}$. Fix $k<n$. ...
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### How to mathematically express the fact that the conditional probability $P(Y|X)$ can be independent of $P(X)$?

Mathematically, $P(Y|X) = \frac{P(X,Y)}{P(X)}$ and so $P(Y|X)$ must depend on $P(X)$. Since $P(Y|X)$ will change when $P(X)$ changes. However, consider this scenario: X = amount of red meat consumed ...
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### Proving Incompleteness of joint sufficient statistic

Let $X_1, ..., X_n$ be a sample from the continuous density $C~exp(-(x-\theta)^4)$ (for $-\infty < x < \infty$) with $\theta$ as unknown parameter. Show that the minimal sufficient statistic is ...
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### How does Variational Autoencoder approximate the joint probability distribution?

I know that in Variational Inference the idea is to approximate the posterior P(z|x, y) and I know that Variational AutoEncoders (VAEs) use the idea of variational inference through neural network ...
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### Gaussian Markov Random Fields - Conditional distribution from jointly gaussian with given precision matrix

Suppose I have jointly normal random vectors $[\bf{v_1}, \ldots, \bf{v_K}]$' with mean $\bf{M}$ and joint block tridiagonal precision matrix $\bf{P}$:  \bf{M}= \begin{bmatrix}\bf{\mu_1} \\\ldots \\...
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### Estimating joint probabilities across two datasets

I would like to estimate the joint probability of two variables from two different surveys conditional on other variables that the two surveys have in common. As an example I'm using data from this ...
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I am having difficulties with the following problem: Assuming $X$ and $Y$ follow a bivariate normal distribution with $\mu = 0$ and $\Sigma=\begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}$ ...