Questions tagged [bivariate]

Concerning two random variables

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28 views

Correlation between nominal and ordinal variables

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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0answers
12 views

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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1answer
34 views

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....
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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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1answer
35 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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3answers
160 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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2answers
243 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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22 views

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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1answer
123 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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1answer
36 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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1answer
106 views

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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53 views

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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1answer
48 views

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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1answer
67 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 ...
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3answers
167 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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1answer
237 views

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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Inverse Distance Weighting for interpolation/gap-filling with bivariate data?

I'm in two minds about this and none of the resources I find online seem to give a clear answer on this. I understand the premise of IDW and have used it before. However, is this method valid (or in ...
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22 views

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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1answer
87 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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1answer
211 views

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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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 ($\...
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1answer
64 views

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 ...
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1answer
63 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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32 views

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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96 views

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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29 views

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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33 views

How to compute the probability of $X_1>aX_2+b$ for a bivariate normal distribution $\mathcal{N}(X_1,X_2)$

Consider a bivariate normal distribution as follows $$ \begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \sim \mathcal{N} \left( \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix} , \begin{pmatrix} \sigma^2_1 ...
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What is the best way to generate fake noncircular data? [duplicate]

How can I generate fake data that resembles this real data from a youtube video: The following is my attempt to do so: ...
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2answers
471 views

Ratio of Two Uniform Random Variables [duplicate]

If X1 X2 are independent Uniform variates on (0,1), Find the distribution of Z=X1/X2. I tried using the CDF method where P(X1<=zX2) is equal to z/2 when z is in(0,1). However, I am unable to find ...
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32 views

Joint continuous probability function with variable bounds

If we have a joint probability density function of the form I've found c to be 3/2 through setting the double integral of the x bounds and y bounds equal to 1, Then I tried to find the marginal ...
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1answer
354 views

Tukey depth intuition

In a bagplot, the inner polygon, called the bag, is constructed using the Tukey depth. What is the intuition of Tukey depth and how is it calculated? A simple two-dimensional example with ten to ...
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23 views

Efficiently comparing bivariate kernel density estimates

I have some bivariate kernel density estimates that I want to compare visually across different regions in space. The main features that I want to highlight are the shapes of the distributions and ...
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13 views

Two stage model where both dependent variables are dichotomous

New here so apologies if I do not explain myself as well as I should. I have survey data of 2 decisions that participants make: the decision to vaccinate themselves (yes/no) and their children (yes/...
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14 views

bivariate conditional joint sensor model

I am struggling to find $P\left( V_t | z \right)$ from $P\left( V_t | z , V_p \right)$. Here $z$ and $V_p$ are independent variables. ...
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1answer
33 views

Bivariate probability inequality into an integral

I am having a hard time understanding one (I assume simple) part of a proof (see the link below if you want the whole thing). Let the two bivariate continuous random vectors: $(X_1,Y_1), (X_2,Y_2)$ ...
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1answer
85 views

Highest probability set and density ratios equal to probability ratios

I came across a pretty result I had not seen before, and wondered if there were more examples For a random variable with an exponential distribution, if you want the highest probability set to ...
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1answer
55 views

Random Sampling from Farlie-Gumbel-Morgenstern bivariate exponential distribution

I would like to obtain an algorithm for generating iid samples from Farlie-Gumbel-Morgenstern bivariate exponential distribution (as described in the book by Johnson and Kotz as Gumbel's Model II ...
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66 views

Time series - measure impact of another time series variable

I am looking for some techniques that would help me measure the (over-time) impact of a variable to another. So let's say we have annual time series data for GDP for 5 countries and I wanted to see ...
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1answer
60 views

Another “not bivariate normally distributed” question

I've found questions very similar to the following. But I haven't found any that involve something of the form $X|Z|$. Let $ Z\sim \mathcal{N}(0,1) $ and $X$ be the discrete random variable such that ...
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82 views

Finding the MGF of a bivariate Normal Distribution [duplicate]

Given ($X$, $Y$) whose MGF is defined as: $$M_{XY}(s, t)=E[e^{sX+tY}]$$ Find $M_{XY}(s, t)$ when $X$ and $Y$ are two jointly normal random variables with $E[X]=\mu_X$, $E[Y]=\mu_Y$, $var(X)=\sigma^...
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2answers
50 views

Correlation between the linear combinations of bivariate normal distributed variables

How can I find the correlation (rho) between $U$ and $V$, Where $U = X_1+X_2$ and $V= X_1-2X_2$ $X_1$ and $X_2$ are normally distributed with $\mu= 1$ and $\sigma= 2$.
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Partial derivative of bivariate cdf

Suppose the bivariate cdf $F(a,b)=Pr(X\leq a, Y\leq b)$ is differentiable in $(a,b)$. Is it true that $\frac{\partial Pr(X\leq a, Y\leq b)}{\partial a}=Pr(X=a,Y\leq b)$?
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2answers
28 views

Relation between bivariate survial function and cumulative density function

I am trying to understand why $Pr(T_1> t_1,T_2 > 2)=1-F_1(t_1)-F_2(t_2)+F_{12}(t_1,t_2)$ My derivation is as follows: \begin{align} Pr(T_1> t_1,T_2 > t_2) &=Pr(T_1> t_1\mid T_2>...
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1answer
111 views

bivariate normal distribution meaning [duplicate]

Does bivariate normal distribution mean the two random variables have normal distributions? is that enough for two random variables to have a bivariate normal distribution or are there some other ...
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1answer
64 views

Multiplying bivariate gaussians by a constant

Say I have the following : $$ (X, Y) \sim N_2(\mu, \Sigma) $$ Then what would be the distribution of $(2X,2Y)$ ? Let $\Sigma = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2\\ \rho\sigma_1\...
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1answer
61 views

How to find a such that X + aY is independent of X-aY for a bivariate distribution [closed]

Suppose $X$ and $Y$ are bivariate normal with equal variance, i.e. $[X, Y] \sim \mathcal{N} (0, \Sigma)$, where $$\Sigma=\begin{bmatrix}1&\rho\\\rho&1\end{bmatrix}$$ Find $a ≥ 0$ such that $X ...
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2answers
105 views

Show the bivariate normal cdf evaluated at (0,0) is increasing in the correlation coefficient

$\begin{bmatrix}\epsilon_{1}\\ \epsilon_{2}\end{bmatrix}\sim N(\begin{bmatrix}0\\0\end{bmatrix},\begin{bmatrix}1,\rho\\ \rho, 1\end{bmatrix})$. Show that the joint cdf evaluated at (0,0), i.e., $F_{\...
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1answer
386 views

Ellipse region shape from standard deviations

I need to draw a bivariate normal distribution ellipse based on this article. It says In the case of the bivariate normal distribution, both approximate and exact methods are available for ...
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32 views

Unbiased estimators of bivariate gaussian means

What are the best unbiased estimators of bivariate gaussian means given covariance matrix? Is there any such estimator that makes explicit use of the covariance matrix, and which is superior by any ...

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