# Questions tagged [bivariate]

Concerning two random variables

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### Expectancy of Euclidean norm of correlated bivariate Gaussian [duplicate]

Consider a noncentral bivariate Gaussian , with nonzero mean, and with ρ not neccesarily zero, and σ1 not neccesarily equal to σ2. I want to find an expression for the expected value of the Euclidian ...
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### 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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### 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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### 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 ...
1 vote
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### 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 ...
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### 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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### 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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### 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
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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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### 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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### python gibbs sampler for bivariate normal distribution, failing to converge

I've been trying to understand Gibbs sampling for some time. Recently, I saw a video that made a good deal of sense. https://www.youtube.com/watch?v=a_08GKWHFWo The author used Gibbs sampling to ...
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1 vote
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### Bivariate basis functions with span invariant to rotation about $z$-axis

Consider the following functions defined over $x,y\in\mathbb{R}$: $f_0(x,y)=1$ $f_1(x,y)=x$ $f_2(x,y)=y$ These functions form a basis with three-dimensional span (the set of all non-vertical planes) ...
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