Questions tagged [bivariate]

Concerning two random variables

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Best fit line of bivariate normal data passes through extrema of level sets

I am learning about the idea of correlation in statistics, and I came across the following Statement: the best fit line of bivariate normal data passes through extrema of level sets. That is, if $(X,Y)...
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Bivariate normal CFD approximation using characteristic function

The normal distribution CFD can be approximated using $$F_X (x)=P[X≤x]=\frac{1}{2}-\frac{1}{π} \int^{\infty}_{0}\operatorname{Re}\left[\frac{e^{-iux}\phi_X (u)}{iu}\right]du$$ where the characteristic ...
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Expression of Kibble's bivariate Gamma distribution PDF

I'm studying the Kibble's bivariate Gamma distribution and found one inconsistency between different papers and I'm not sure which is correct. In the Smith et al. 1982, A bivariate Gamma Probability ...
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1 vote
1 answer
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Erroneous Argument for uncorrelated implies Independence

I've been working on the problem where for a bivariate normal random variable (X,Y), uncorrelated implies Independence. However, I realized that I didn't use the bivariate normal assumption, so there ...
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1 answer
68 views

From univariate to joint convergence in distribution

Let $X_n \rightarrow_d X$ and $Y_n \rightarrow_d Y$ where $X$ and $Y$ are i.i.d standard exponential random variables. However, I do not have that for any $n$, $X_n$ and $Y_n$ are independent. Can I ...
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1 vote
1 answer
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Estimating the cumulative probability of a bivariate normal distribution

I have a quick question regarding working out the probabilities of a bivariate normal distribution. To my knowledge, there is no nice closed-form for a cumulative distribution function for the ...
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2 votes
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A question about product and division of random variables

Taken three real continuous random variables $X,Y,Z$, non negative (to simplify the handling of the problem), with respective pdf's $p_X , p_Y , p_Z$ . Then if $$ Z = XY\quad \left| {X,Y\;indep.} \...
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1 vote
1 answer
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How to find the distribution of a function of two random variables?

I have the joint pdf $$f(x_1,x_2)=\begin{cases}\frac{1}{4}\;,\ -1<x_1,x_2<1\\ 0\;,\ \text{otherwise.}\end{cases}$$ I have to find the distribution of $Y=\frac{X_2}{1+X_1^2}.$ I have tried to ...
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1 vote
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43 views

Test for bivariate normality assumption

Assume two time series $X_1$ and $X_2$, where $X_i=(x_{i,1},...,x_{i,T})$. How can we test the assumption of bivariate normality for these time series? (Assume each of the time series are stationary ...
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3 votes
1 answer
111 views

How can population variance be estimated from a bivariate sample?

Let's assume a bivariate population with a correlation $\rho$ and a common $\sigma$ so that $\Sigma = \sigma^2 \begin{pmatrix}1 & \rho \\ \rho & 1\end{pmatrix}$. I would like to know the ...
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28 views

Fitting bivariate normal distribution with covariate-dependent correlation parameters

Suppose we fit $$ \begin{bmatrix} X \\ Y \end{bmatrix} \sim \mathcal{N} \left( \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 & \rho(C) ...
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Is it possible to calculate a bivariate normal density using only the univariate normal density function? [duplicate]

Suppose $(X,Y)$ are two jointly normally distributed random variables. Suppose further that we want to calculate the density of $(X=x,Y=y)$. Is it possible to calculate this density if we do not have ...
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Fitting a bivariate logit model in R using binom2.or and model diagnostics?

I took the advice from the comments and searched for bivariate probit models. But new questions arose regarding model diagnostics and evaluation. First my original question: Ways to analyze (related) ...
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Two possible definitions of confidence regions: which one to choose?

Let's say you have a parameter vector $(p,q)$ consisting of two proportions, and you want to find a confidence region for the estimator $(\hat{p},\hat{q})$. Define \begin{equation} H(x,y,p,q)=\frac{n}{...
2 votes
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Is a bivariate copula relevant in this physics setting manifesting uniform univariate marginals--and, if so, how can it be constructed?

To quickly place our probabilistic (copula) question in its subject matter setting, we note that a fundamental concept in quantum theory is that of entanglement QuantumEntanglement. The states of ...
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What are examples of symmetric copulas $f(x,y)=f(y,x)$ having relative minima for $f(x,x)$?

In a previous posting on this site RepulsiveBehavior I attempted to detail a quantum-information-theoretic separability/entanglement problem I am pursuing. Detailed issues of sampling sizes for a data ...
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Do any standard copulas fit well these sampled bivariate data--exhibiting repulsive behavior--having uniform marginals

I'm currently developing a data set that consists of two $50 \times 50$ matrices, which I designate as q1 and Q1. I strongly believe (bordering on formal proof [cf. Corollary 1 in marginalinvariance]) ...
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Calculate chance of radius being smaller then X in a bivariate normal distribution

If I have two variables X and Y that are both normally distributed with the same standard deviation σ and the same mean 0. They come together to form a circular bivariate normal distribution. How can ...
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Your class has 100 students and they have 5 elective courses to choose from. In each course, the proportion of students is equal in population

I am actually unable to understand the question and would appreciate it if someone can help with that. For the above problem statement, there are three questions that I need to answer Make a ...
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Bivariate Distribution for Milstein Scheme

I am seeking a density for $$Y_{t+1}\mid Y_{t},$$ where $$Y_{t}=\begin{pmatrix}Y_{t}^1\\ Y_{t}^2 \end{pmatrix},$$ and for $i=1,2$, the $i$th component of the 2-dimensional Milstein scheme is given ...
1 vote
1 answer
61 views

Covariance between partitions of a normal distribution

A bit of a contrived example, but if I had a sample of $X_1,\dots,X_n \stackrel{iid}{\sim} N(\mu,\sigma^2)$ (in this case $\mu$ is unknown but $\sigma^2$ is known), and then calculated the arithmetic ...
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1 vote
1 answer
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Derivative of a Bivariate normal CDF with respect to its variables

Following up on the question (and answers) here, I'm trying to derive $\frac{\partial \Phi(x_1, x_2|\mathbf{\underline{\theta}})}{\partial x_1}$ and $\frac{\partial \Phi(x_1, x_2|\mathbf{\underline{\...
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Multivariate Analysis and AOR

Hello All, I understand how we can calculate the Bivariate analyses using the table 3 in table 4. But can anyone please help me understand how to calculate Multivariate analyses using AOR. Are there ...
1 vote
0 answers
87 views

Does an R-based implementation of Fasano and Franceschini's (1987) 2D Kolmogorov-Smirnov test exist? [closed]

First I would like to recognize that similar versions of this question have been asked before, however these either did not pertain to Fasano and Franceschini's (1987) modification of the 2D KS test ...
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Let X,Y be bivariate normal , what is E[X|Z] where Z = X + Y? [duplicate]

I am trying to understand how does expectation and variance looks when Let X,Y be bivariate normal I want understand E[X|Z] and Var[X|Z] when Z = X + Y
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What's the distribution of the euclidean distance between a fixed point a bivariate normal distribution?

I have a problem where I'm interested in the distance between a fixed point and a bivariate normal distribution (where the two random variables are correlated). How does one find the distribution of ...
1 vote
1 answer
147 views

Correlation between nominal and ordinal variables [duplicate]

In my survey data I have two variables: One is an ordinal variable with 5-scale scoring from Agree to Disagree. My second variable is an nominal variable where the participants had to choose from 7 ...
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Find skewness and kurtosis of a bivariate scatterplot

I was reading this paper in the medical field which proposes a method using two radiological parameters in order to distinguish benign and malignant lesions. In their paper the Authors show this plot ...
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1 answer
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Joint entropy of a bivariate Gamma probability density function

In Nadarajah and Kotz, 2009 (https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-39/issue-1/Four-Bivariate-Distributions-with-Gamma-Type-Marginals/10.1216/RMJ-2009-39-1-231....
2 votes
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Covariance in Robot Pose from Covariance in Line Segments

The solution to this might be completely obvious, so I apologize beforehand. The purpose of this question is for me to get better intuition on how to use covariance matrices of correlated variables in ...
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1 answer
119 views

Proving that a random vector is not bivariate normal

Suppose X,Y are random variables and their joint pdf is given by: f(x,y)=2g(x)g(y) where x*y>0, and zero otherwise. g(x) and g(y) are pdfs of standard normal distribution. I was first able to prove ...
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2 votes
3 answers
239 views

How to find the variance(s) of a bivariate normal density such that 95% of the mass is within a certain radius from the mean defined by a point A?

I would like to find the variance of a bivariate normal density (BND), centered at the mean M, such that 95% of its mass is within a certain radius, which depends on the position of a point, A. (Note: ...
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4 votes
2 answers
832 views

EM algorithm for MLE from a bivariate normal sample with missing data: Stuck on M-step

I'm trying to understand applying the EM algorithm to compute the MLE in a missing data problem. Specifically, suppose $(x_1,y_1),\ldots,(x_n,y_n)$ is a random sample from the bivariate normal ...
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Mean of uniform two-dimensional probability density function

I am trying to calculate the mean of a two-dimensional probability density function, which looks like: and is defined by I know that I can calculate this by However, this is where I get stuck, as I ...
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3 votes
1 answer
363 views

Linear least-square fitting of two variables with uncertainty on both

I am trying to find an R function to calculate the linear least-square fitting of two variables when both have an error (expressed as standard deviation). I have found this problem referred to in half ...
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6 votes
2 answers
183 views

"Information" Correlation

(Let $X$ and $Y$ be random variables, sufficiently nice for my question to make sense.) $$ \text{Correlation} $$ $$ \rho(X, Y) = \dfrac{\text{cov}(X, Y)}{\sqrt{\text{var}(X)}\sqrt{\text{var}(Y)}} $$ ...
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plotting bivariate normal distribution samples

I have generated two samples $\underline{X_i}$ and $\underline{Z_i}$ $\underline{X_1}$ $\underline{X_2}\dots \underline{X_{5000}}$ , while $\underline{X_i} \sim N_2[(1,2)^T,\begin{pmatrix}2&1.5\\ ...
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2 votes
1 answer
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Bivariate Random Transformation finding CDF

Problem Assume $Y_i, i=1,2$ are independent with pdf-s $$f(y_i;\theta)=\frac{1}{\theta}e^{\frac{-y_i}{\theta}} \forall y_i>0, \, \theta >0$$ Let $Y = Y_1 + Y_2$, and show that $$F(Y) = P(Y \le y)...
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-1 votes
2 answers
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Why do elliptical copula densities contain $x_1$ and $x_2$, but Archimedean copula densities contain $u_1$ and $u_2$?

$$c\left(u_{1}, u_{2}\right)=\frac{1}{\sqrt{1-\rho_{12}^{2}}} \exp \left\{-\frac{\rho_{12}^{2}\left(x_{1}^{2}+x_{2}^{2}\right)-2 \rho_{12} x_{1} x_{2}}{2\left(1-\rho_{12}^{2}\right)}\right\}$$ is the ...
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0 votes
1 answer
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multivariate seasonal time series in dlm in r

I am trying to build a dynamic linear model in R for my bivariate seasonal (monthly ) time series. I found the following resources which help me to model bivariate cases but there is no seasonality ...
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1 vote
1 answer
884 views

Relation between tail-dependence and correlation in t-copula

I have data for two variables, say X and Y and I fit a bivariate t-copula to this data. The fitting of a t-copula gave me two values: a degree-of-freedom (nu = 4.5) and a correlation matrix. From this ...
3 votes
3 answers
2k views

Expected value of a bivariate distribution as an integral

Let's assume an absolutely continuous random variable, $X$, with PDF $f(x)$. $$\mathbb{E}\big[X\big] = \int_{\mathbb{R}}xf(x)dx$$ If $X\sim f(x_1,x_2)$ is multivariate, then it makes sense to me to ...
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3 votes
1 answer
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Sample correlation is also a MLE estimator

On page 599 of this book, the author states (without proving) that for random samples $(X_1, Y_1)$, ..., $(X_n, Y_n)$ from a bivariate normal distribution, the sample correlation coefficient \begin{...
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Analytically solving the conditional normal distribution [duplicate]

I'm trying to derive the gaussian conditional distribution for a 2 variable gaussian. I'm doing this as I'm studying Gibbs and need to learn how to derive conditionals the analytic way; practice makes ...
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3 votes
1 answer
98 views

Multivariate gaussian bivariate gaussian proof

I'm having trouble seeing how the multivariate gaussian formula evaluates to the bivariate gaussian. See multivariate PDF, source: http://cs229.stanford.edu/section/gaussians.pdf [![multivariate][1]][...
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1 vote
1 answer
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How to derive OLS estimator for binary or bivariate regressor in terms of population moments

I am wondering how to find an OLS estimator for beta_1 in terms of population moments when the corresponding independent variable is a Bernoulli random variable equal to either 0 or 1. My professor ...
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2 votes
1 answer
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How do I generate a confidence region for a set of sample from a bivariate posterior?

I have a set of samples generated from a posterior function as shown below: I want to generate a bivariate High Posterior Density (HPD) credible region for the samples as in the below example ($\...
0 votes
1 answer
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Conditional Probability Uniform Bivariate Transformation Distribution

I'm reviewing probability theory from years ago and am a bit rusty. I'm not sure how to calculate the conditional probability for a uniform distribution after a bivariate transformation. Suppose X and ...
2 votes
1 answer
130 views

Does $(X,X)'$ follow a bivariate normal distribution?

I'm fairly new to multivariate distributions. I'm trying to figure out if $(X,X)^{'}$ follow a bivariate normal distribution (the prime = transposed). If $X\sim N(\mu, \sigma^{2})$ where $\mu \in \...
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To prove that a set is a subspace

I am having trouble showing that the criteria for a set to be a subspace is true for this particular set. Using the condition given here , I can obtain correlation in terms of $b$ and $a$ but I am ...
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