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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6
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72 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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266 views

What is the density of a markov chain when its transition probabilities have densities with respect to different measures?

I have a homogenous, discrete time Markov process, $(X_n)_{n\geq 0}$, with state space $\mathbb R_+$. Its transition probabilities have a density, $f(x_n\mid x_{n-1})$, with respect to the measure $\...
5
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0answers
134 views

Joint distribution of column sums when row sums are fixed

Suppose I have an $m$ by $n$ table $X_{ij} \in \{0,1\}$, where in each row, $r$ randomly chosen entries are set to 1 (the rest are 0), i.e. $\sum_j X_{ij}=r$. I know that e.g. the column sum $\sum_i ...
5
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0answers
246 views

Books for mixed distributions (continuous and discrete)?

What is a good book that covers mixed distributions? Most statistics books either only briefly mention them or do not cover the topic at all. I'd like to have a comprehensive resource covering ...
4
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182 views

Why does test on Pearson correlation require bivariate normality?

For a pair of random variables $X$ and $Y$, we can compute their Pearson correlation coefficient $r$ and conduct hypothesis testing on the null hypothesis $H_{0}:r=0$ with the $t$ statistic $t=r\sqrt{...
4
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0answers
749 views

directed bayesian network and factor graphs

I have a directed bayesian given by the figure below. In the figure the circles are random variables and the shaded ones are observed. The rectangular nodes are constants representing the hyper ...
4
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0answers
3k views

Generating samples from Copula in R

Suppose I want to model dependence between $d$ r.v.´s $Y_1,...,Y_d$ with the copula $C_\theta$, where $\theta$ are the corresponding parameters of that copula. I've also determined the correlation ...
4
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424 views

Reconstructing joint distribution from marginals

I think this is a rather open question. Suppose I have bi-dimensional data $(x_i, y_i)$. I have some reasonable model for the marginals, say distributions $F_X$ and $F_Y$ (parametric). How to ...
4
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0answers
432 views

How to use an initial posterior for recursive / sequential updating in WinBUGS

I am using WinBUGS to estimate / update the parameters of a model. The model is: $$ \begin{aligned} D(T,B,a)&= B*(a_0+a_1T+a_2T^2+a_3T^3)+error(B,T,a) \\ error &= \mathcal N(0, B^{0.5}a_4(...
4
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1k views

Deriving conditional distribution using Gaussian copula

This question shows how to derive an analytical expression for the conditional distribution from a multivariate normal. I am curious how well this extends to when there's a Gaussian copula, but ...
3
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72 views

Joint distribution of log hazard ratio estimates for two outcomes

I have a study where individuals are randomized to a treatment or control group, and there are two time-to-event outcomes $S_i$ and $T_i$ measured for each individual $i$. The two outcomes have some ...
3
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37 views

When (if ever) is the sum of two dependent geometric RVs negative binominal?

Imagine you have two random variables $X $ and $Y$, you know $$ X \sim \text{Geometric}(p) \\ X + Y \sim \text{Negative Binomial}(2, p) $$ I am interested in what if anything can be said about the ...
3
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1answer
143 views

When $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim(X_0-X_1, X_0-X_2)$?

Consider a bivariate distribution function $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question: Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that $$...
3
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1answer
815 views

Conditional Probability vs Conditional Probability Distribution

I am having hard time interpreting the relationship, if any, between conditional probabilities vs. conditional probability distributions, in particular, regarding the number of random variables ...
3
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0answers
122 views

Prove that the joint density of independent multivariate normal variables is a matrix-normal

Let $X_1,...,X_n \sim N_p(\mu_i,\Sigma_i)$ be Multivariate Normal a.v. independent. Show that $W = (X_1,...,X_n) \sim MN(M,\mathbb{I},\Sigma)$ where $M = [\mu_1 \mu_2...\mu_n]$ and $\mathbb{I}$ ...
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2k views

Probability Density Function of a linear combination of 2 dependent random variables, when joint density is known

Let's say there are two dependent random variables $X$ and $Y$ with joint density function $f$. What is the PDF of the weighted sum of these two variables, $Z = aX + bY$? Thanks in advance for any ...
3
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277 views

Pattern-mixture models

I am currently looking at pattern-mixture models but I don't seem to understand them and I wonder whether someone could help. I can see the model comes from the factorisation $ f(y,r;\phi, \theta)=...
3
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1answer
2k views

Probability that one random variable is larger than another with known correlation

Let's say I have a normally distributed random variable $X_1$ with known standard deviation $\sigma_1$ and $E[X_1]$ is $0$. Let's say I have another variable with known standard deviation $\sigma_2$ ...
3
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104 views

Calculate the probability mass function of the new random variable $Z≡X+Y$

I am having trouble with these. How do I complete the table? Consider the random variables $X$ and $Y$ with joint distribution as given below. ...
3
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344 views

Finding a test statistic when you don't know the distribution?

I am working on this problem from my class and it has stumped me for a while now. I will show a picture of the problem and then my work/thoughts: Now we are not given the individual distribution of $...
3
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0answers
715 views

Bayesian update with multiple parameters

In the past I have been able to do Bayesian updating when there is just one parameter which I am trying to estimate. I know a bit about Bayesian methods but I am confused by how to extend them to ...
3
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0answers
120 views

joint distribution, probability, calculating probabilities under false independent assumption, when the random variables are actually dependent

Suppose I have random variables $X,Y,Z$ and I would like to compute the probability that random variable $X$ is smaller than $Y$ and $Z$: $$ \pi_X \overset{def}{=} Pr(X < Y, X < Z) = \int Pr(x ...
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1k views

Non-parametric estimate of conditional expectation

I have a (fairly smooth) function $f$ and a sample $\{(x_i,y_i)\}_{i=1,\ldots,N}$ from the joint distribution of the random variables $X$ and $Y$. I would like to estimate the conditional expectation ...
2
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22 views

Confusion about two Gaussian distributions

From here, it says that, linear combination of two Gaussian distribution, are always Gaussians. However, Let 𝑋 be standard normal and 𝜀=±1 with probability 1/2 each, independently of 𝑋. Let 𝑌=𝜀𝑋...
2
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1answer
42 views

How to show two variables are asymptotically independent

Let $X_1,...,X_n$ be iid from $Exp(\theta)$ with density function $f(x) = \frac{1}{\theta}e^{-x/\theta}$. Show that $M_n = X_{n:n} - \theta \ln(n)$ and $T_n = nX_{1:n}$ are asmyptoically independent ...
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25 views

Joint Entropy of two time series

I have two time series of same length like following: ...
2
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0answers
16 views

If the joint density $f_{X_1,…,X_n}(x_1,…,x_n)$ is symmetric about the origin, does this imply that each marginal cdf $F_{X_i}(0)=1/2$?

If the joint density $f_{X_1,...,X_n}(x_1,...,x_n)$ is symmetric about the origin in the sense that for any $(x_1,...,x_n)$, it holds that $f_{X_1,...,X_n}(x_1,...,x_n)=f_{X_1,...,X_n}(-x_1,...,-x_n)$ ...
2
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0answers
23 views

distribution of a multivariate transformation with different dimensions

Given a continuous multivariate random variable $\bf X$ with pdf $f_\mathbf X$, and an invertible transformation $g:\mathbb{R}^n \to \mathbb{R}^n$, it is known that the joint pdf of $\mathbf{Y}=g(\...
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12 views

Best methods to integrate summed probability and uncertainty from multiple sources?

I am trying to calculate risk of coastal flooding. In our model, we have 4 major sources of uncertainty: Elevation at location (height + gaussian error) Sea level rise (value at any given year + ...
2
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0answers
90 views

The joint distribution of two different sums of the same independent uniform random variables

Given $k$ independent, uniform random variables $X_i \sim U(-a,a)$, $i=0,\dots k$, and two sets of coefficients $\{\alpha_i\}$ and $\{ \beta_i\}$, let $U = \sum_{i=1}^k \alpha_i X_i$ and $V = \sum_{i=...
2
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0answers
64 views

Combining subjoint distributions to create a larger joint distribution

I am trying to construct large joint distributions through smaller joint distributions and I'm not sure how to approach the literature. I am curious if there exists a function which can take n ...
2
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1answer
115 views

Visualizing separability / independence

I’d like to visually ‘see’ the independence of random variables. I tried plotting f(x), f(y), and f(x, y) for independent and dependent pairs of variables. However, the difference is still not ...
2
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0answers
47 views

How is exchangeability related to covariate shift?

I understand that exchangeability refers to the notion that the order of data in a sequence does not affect the joint distribution of that data. In a sense, the current data we possess is from the ...
2
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0answers
90 views

Jensen–Shannon divergence relation between marginal and joint

I have the following distributions- Joints: $p_1$(x,y) = $\rho_1$(x) $q_1$(y|x), $p_2$(x,y) = $\rho_2$(x) $q_2$(y|x) And the marginals: $\rho_1$(x), $\rho_2$(x). Is it possible to provide any ...
2
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0answers
100 views

Check whether a random sample comes from an elliptical distribution?

How can I check whether it is a reasonable assumption to say that a multivariate sample $x_1,...,x_n$ comes from an elliptical distribution, such as a normal distribution or a t-distribution? In the ...
2
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0answers
102 views

Estimating parameters and simulating a multivariate time series (O-U?) process

Say I have 5 time series that may or may not be correlated with each other. I also suspect they individually may be mean reverting. I'd like to 1) be able to test the hypothesis they are ...
2
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0answers
36 views

Exploiting the joint density when averaging measurements

I have two measurements of a quantity from two different sensors, each with their own variance based on the sensors' specifications, etc. From collecting a lot of training data, I have access to the ...
2
votes
1answer
58 views

Estimating the likelihood of an observed path for a continuous state space Markov process

Problem statement Let $\mathbb{X} \subset \mathbb{R}^{k}$ and let $p:\mathbb{X}\times\mathbb{X}\rightarrow\mathbb{R}$ be a density kernel on $\mathbb{X}$. We assume the following model for the ...
2
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0answers
210 views

Classification in the presence of dependent events

In a classification problem, I have a number of $n$ disjoint and exhaustive (and also likely) events $C_i$ defined as a joint event by $$ C_i = H_{i1} \cap H_{i2} \cap ... \cap H_{im}, $$ where $H_{...
2
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0answers
315 views

Conditional cdf

I want to know that how conditioning will affect the CDF of dependent random variables. More specifically, let's suppose, $\Gamma_R={g\over A}$ and $\Gamma_D={g\cdot h\over B}$, where $g$ and $h$ are ...
2
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0answers
948 views

Maximum likelihood estimation (MLE) for a Bayesian network with binary variables

I have below Bayesian network of 4 variables A, B, C, and D. All variables are binary except D, which is real-valued. $P(A|B,C)$ is defined by the conditional probability table (CPT) on the right. ...
2
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0answers
169 views

Express the density of a function of two random variables using the Gradient and the joint density

I would like to know if it is possible to express the density $f_Z(z)$ of a function $Z = g(X,Y)$ of two continuous "nice" random variables $X$ and $Y$ only using the joint density $f_{XY}(x,y)$ and ...
2
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0answers
42 views

How to find $\mathbb{P}(X<C)$ where both $X$ and $C$ are RVs and $X$ depends on another RV?

I have a scenario like the following: $C$ is a random variable that follows $\operatorname{Uniform}(0, 3.5)$. $T$ is a random variable with $\operatorname{Exponential}(\operatorname{mean}=3)$ for ...
2
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0answers
124 views

Using copulas to fit hourly observations to daily data

I've been trying to automatically find low periods in some data that I have. The data is structured as hourly observations across a period of two years. Thus far I've experimented with a number of ...
2
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0answers
113 views

should the product of 2 independent binomials converge to a normal distribution for large sample sizes?

I have been studying 2x2 contingency tables and specifically I have been looking at situations where the marginals are fixed by design for one categorical variable. As an example, suppose a ...
2
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0answers
28 views

Independence of ordered statistics part 2

In relation to my previous question, I have a second problem. How to show that $U_1+U_2$ is independent of $Y_3$? I can find out the pdf of $U_1+U_2$, as well as of $Y_3$, however please help me ...
2
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0answers
30 views

What are the values of output $g_i$ in Bengio's paper “Taking on the Curse of Dimensionality in Joint Distributions Using Neural Networks”

Figure 2 of Bengio's paper "Taking on the Curse of Dimensionality in Joint Distributions Using Neural Networks" describes a neural network structure for estimating a joint probability distribution. ...
2
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0answers
203 views

how to find asymptotic joint distribution of two linear combination of order statistics?

Suppose I have n order statistics from some unknown continuous distribution funciton F(x), $X_{1}\leqslant X_{2}\leqslant...\leqslant X_{n}$. And I have two linear combination of these order ...
2
votes
2answers
565 views

Can I compare Mahalanobis distances from different distributions?

I have a multivariate dataset representing multiple locations, each of which has a set of reference observations and a single test observation. For each location, I would like to measure how anomalous ...
2
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0answers
56 views

Calculate mutual information of twitter's users interactions

I have a research project about finding out user's interaction on Twitter. If I want to calculate the mutual information between two users A,B: I(A,B), to know the probability of A and B appearing in ...

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