Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit 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.

5
votes
0answers
188 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
votes
0answers
126 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 ...
4
votes
0answers
117 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
votes
0answers
101 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 $\mathbb{I}$ ...
4
votes
0answers
111 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 ...
4
votes
0answers
523 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
votes
0answers
401 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(...
3
votes
0answers
320 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 \...
3
votes
0answers
108 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 ...
3
votes
0answers
98 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
votes
0answers
236 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
votes
0answers
549 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
votes
0answers
116 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 ...
3
votes
0answers
2k 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 ...
3
votes
0answers
256 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 ...
3
votes
0answers
854 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 ...
3
votes
0answers
57 views

How to find the connection between variables

Assume that you want to approximate your joint distribution on set of variables via decomposing it into several smaller distributions. Assume that you don't know any prior knowledge about the ...
3
votes
0answers
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
votes
0answers
208 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 ...
2
votes
0answers
40 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
votes
0answers
65 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
votes
0answers
199 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
votes
0answers
156 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
votes
0answers
738 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
votes
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
votes
0answers
101 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
votes
0answers
25 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
votes
0answers
28 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
votes
0answers
133 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
0answers
228 views

Distribution of ratio of two independent normal variables

My objective is to find out the distribution of $A/B$ given $A \sim N(a,b); B \sim N(c,d)$. I set $Z_1$ equal to $\frac{(A-a)}{\sqrt{b}}$ and $Z_2$ equals $\frac{(B-c)}{\sqrt{d}}$ such that $Z_1$ ...
2
votes
0answers
54 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 ...
2
votes
0answers
87 views

Estimating a joint distribution from observed max and min samples

Suppose that you have jointly distributed $N$ (~100) random variables, $\{X_1,\ldots,X_N\}$, and this distribution is unknown to you. However you do know that their sum is zero by construction. Having ...
2
votes
0answers
191 views

Does relative Kullback-Leibler divergence exist?

Suppose I have two multivariate normal distributions. I have computed the KL divergence ($d_{KL}(N_1, N_2)$). Is there a way to measure a relative divergence between these two distributions? For ...
2
votes
0answers
1k views

The formula for covariance in terms of joint cdf

I want to show that $$\newcommand{\cov}{\operatorname{cov}}\newcommand{\d}{\mathrm{d}}\cov(x,y) = \iint (F_{X,Y}(x,y) - F_X(x)F_Y(y))\,\d x\,\d y$$ However, I have no idea how to start. I know that $...
1
vote
0answers
24 views

Expected function evaluation of random variable w.r.t. different distribution

Suppose I have two continuous random variables on the same domain, $\xi \sim \mathbb{P}, \xi' \sim \mathbb{Q}, \in \Xi$ and joint probability $(\xi, \xi') \sim \Pi \in \Xi^2$ . Now I would like to ...
1
vote
0answers
28 views

How to evaluate double Integral with importance sampling

I am trying to recreate the Bayesian Hierarchical Clustering algorithm using Python. The example in section two requires evaluating the following double integral (univariate case): \begin{align} p(...
1
vote
0answers
18 views

The joint distribution of Y=AX and Z=BX given a projection matrix A and residual maker matrix B, and a random vector X with known pdf?

This question follows on from a previous question I asked which was answered. It turns out my question lacked some important details, which was revealed by the answer posted on that thread. This is ...
1
vote
0answers
11 views

create joint prob distribution or empirical relation for two variables

There are two variables, X1 and X2. The experimental study shows that they are highly correlated. Are there any reliable ways to create an empirical mapping(or equation) between X1 and X2. Assuming ...
1
vote
0answers
14 views

identifying which of $d$ normal distribution generated a given sample

I have $d$ Normal Distributions, $N_1(\mu_1, \sigma_1^2) \cdots N_d(\mu_d, \sigma_d^2)$. We pick one of the $d$ distributions with each distribution having a probability of $\frac{1}{d}$ of being ...
1
vote
0answers
10 views

Line of Best Fit

If we have a dataset with two variables, X & Y, we can find the line of best fit using the empirical data (and whatever method suits you best). However, what if know the true joint distribution ...
1
vote
0answers
51 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 ...
1
vote
0answers
30 views

Optimize an objective based on a trained model

I want to find a joint optimal subset based on individual scores from a predictive model. Example: Say I have a set of customers and a set of products. And I have trained a model for predicting the ...
1
vote
0answers
14 views

Probability of successful match with two faces in image

A photo containing faces of two different people is compared to labeled images of faces in database. The probability of a match on the first person is .70. The probability of a match on the second ...
1
vote
0answers
35 views

Joint Probability Distribution and covariance

If $$f(x,y)=1/4 $$ $x=-3,y=-5; x=-1,y=-1; x=1,y=1; x=3,y=5. $Find cov (x,y). I know the formula for cov (X,Y) but I'm stuck at finding E (x) and E (y).
1
vote
0answers
42 views

Resample from data with constraints to the marginal distribution

Motivation This problem comes from the situation where I have a non-random sample of individuals for which $p$ variables are measured. The target is to extract a subset of individuals which would be ...
1
vote
0answers
34 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 ...
1
vote
0answers
75 views

Joint probability distribution of geometric distribution

Let $X$ and $Y$ be independent and identically distributed $(i.i.d.)$ r.v.’s, each having the probability distribution, $p(k) = (1 − λ)λ^k$; $k = 0,1,...$ where $λ :(0; 1)$ is a constant. Define $U = ...
1
vote
0answers
66 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 ...
1
vote
0answers
32 views

marginalization of joint distributions

I am trying to understand the following sentence, section 2.2, in this paper: "...it is required that the joint mode $p(x,z,a)$ gives back the original $p(x,z)$ under marginalization over $a$, thus $...
1
vote
0answers
50 views

Factorization of product distribution

I would like to obtain a factorization of the probability distribution of a random variable $Z$ which is the product of two independent random variables $\delta=\varepsilon X$ where $\varepsilon \sim ...