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1answer
24 views

Which error is displayed in an error ellipse?

I have some bivariate data and I have calculated the error ellipse in the following way: I have first calculated the covariance matrix and then to obtain the radii of the ellipse I have taken the ...
2
votes
2answers
42 views

Bivariate Normal Distribution Mean

$X$ and $Y$ have bivariate normal distribution and have joint pdf \begin{equation*} f\left( x,y\right) =a\exp \left( \frac{-1}{2}\omega \right) ,\text{where }% \omega =6x^{2}+12y^{2}-16xy-8x+24 ...
1
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0answers
11 views

Many forms of bivariate distributions

Why do we have many forms of bivariate distributions? An example: Bivariate Exponential Distribution. I understand that they have been derived from transformations, conditions on marginals or ...
1
vote
1answer
20 views

How to compute for Bivariate Logistic Distribution

This is the logistic distribution of single random variable (taken from Wikipedia). $x$ = random variable $\mu$ = mean of all random variables $s$ = variance. Now, I want to do a Bivariate ...
0
votes
1answer
28 views

Probability of object collision

I've been searching for a few days on a number of sites but I can't seem to find a good answer for this. I'm developing a collision detection program using Unscented Kalman Filter and predicting ...
2
votes
0answers
25 views

How to compare 37 dichotomous variables (lab tests) to 31 dichotomous variables (diagnosis)

I've constructed a database based on chart review of patients. Patients had a variety of tests performed (37 different types, which were either positive, negative, or not performed aka 1/0/blank - I ...
2
votes
1answer
96 views

Confidence interval for distance from center

We have a bivariate normal process where $X \sim N(\mu_x, \sigma), \, Y \sim N(\mu_y, \sigma)$, with no covariance. $(\mu_x, \mu_y)$ are unknown. (For convenience we can assert that $\sigma = 1$, or ...
2
votes
1answer
36 views

inequality in bivariate normal variable

Let $U_1=(X_1,Y_1)^T,\dots,U=(X_n,Y_n)^T$ are i.i.d. copies of $U=(X,Y)^T\sim N_2(0,\Sigma)$ where $$ \Sigma= \begin{pmatrix} \sigma^2 & \rho\sigma\tau \\ \rho\sigma\tau & \tau^2 ...
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0answers
10 views

Increasing function defined as the covariance bivariate normal

Suppose $c>0,\sigma>0$ and $\tau>0$ are fixed real constants. Then I'd like to prove that the function $g_c:(-1,1)\mapsto\mathbb{R}$ defined by \begin{equation} ...
2
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0answers
27 views

Bivariate One-Sided Chebyshev Inequality (Symmetric Case)

Let $X$ and $Y$ be random variables with finite means $\mu_X$ and $\mu_Y,$ finite variances $\sigma_X^2$ and $\sigma_Y^2,$ and correlation $\rho.$ Let $A$ be the event that $X \leq \mu_X + k\sigma_X$ ...
0
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0answers
13 views

Proof for bivariate conditional mean of Gaussian dist [duplicate]

I see that a lot of questions are answered here for multivariate and bivariate conditional distributions. But I did not find the proof of these equations (I need just for bivariate case). to get ...
0
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0answers
20 views

Distance from bivariate Gaussian mean in terms of variance

Not sure if my question is a valid one but I will just put it out here. Consider a bivariate data set $(x_i, y_i)$ $[i=1,...,n]$ to which a bivariate Gaussian Distribution is fitted. Now, consider ...
8
votes
3answers
198 views

Where is density estimation useful?

After going through some slightly terse mathematics, I think I have a slight intuition of kernel density estimation. But I am also aware that estimating multivariate density for more than three ...
3
votes
2answers
98 views

If any two variables follow a normal bivariate distribution does it also have a multivariate normal distribution?

Bivariate and multivariate distribution relationship. If we have say 3 variables where any two variables follow a normal bivariate distribution, then does it necessarily follow a multivariate ...
0
votes
1answer
100 views

Which Venn diagram is appropriate here for statistically independent, uncorrelated and orthogonal random variables?

I understand concepts more with visualizations. So I made a Venn diagram for statistically independent, uncorrelated and orthogonal random variables. But I am in a confusion which of the below Venn ...
0
votes
1answer
37 views

Is there a formula for the standard error of both the explanatory and response variables

I was reading a social science paper that tried to explain the correlation between two variables. In that reference there was mention of its standard-error and p-value and that got me to thinking ...
4
votes
1answer
63 views

CDF for uncorrelated bivariate normal

I have an uncorrelated bivariate normal process: $X \sim N(0, \sigma_1), Y \sim N(0, \sigma_2), \rho = 0$, and I'm interested in the distribution of $Z = \sqrt{X^2 + Y^2}$. I know that if I'm willing ...
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0answers
32 views

Generating null distributions by a residual permutation procedure

I am trying to understand the method described in this paper which describes an hypothesis-testing framework for stable isotope ratios. The data are in a bivariate isotopic space and the metrics that ...
2
votes
1answer
63 views

How to create a bivariate normal distribution

I have a data set like this: df Income Education_in_years 40,000 10 50,000 9 70,000 12 30,000 5 100,000 20 I would like to create a ...
2
votes
1answer
92 views

Find the distribution of (X, X+Y) when X and Y have a given joint Normal distribution

Let random variables $X$ and $Y$ be independent Normal with distributions $N(\mu_{1},\sigma_{1}^2)$ and $N(\mu_{2},\sigma_{2}^{2})$. Show that the distribution of $(X,X+Y)$ is bivariate Normal with ...
0
votes
1answer
45 views

Constructing a bivariate distribution from two gamma-distributed random variables with nonlinear dependence?

I've got 2 gamma-distributed random variables $(X,Y)$ with arbitrary scale and shape parameters. Further, $Y$ should be a non-linear function of $X$, lets say $Y=\sqrt{X}$. What I am interested in is ...
2
votes
1answer
91 views

Bivariate normal probabilities

Let $(X,Y)\sim N(\mu_x=1,\mu_y=1,\sigma^2_x=4,\sigma^2_y=1,\rho=1/2)$. Compute $P(X+2Y\leq 4)$. How do you compute probabilities of a bivariate normal? For a regular normal distribution I remember we ...
0
votes
1answer
206 views

Bivariate Gaussian distribution test in R

Regarding the assumption that $X$ and $Y$ need to be normally jointly distributed, when applying Pearson's correlation test... what is a good test (function and package) in R to accomplish this task? ...
4
votes
2answers
116 views

Two normal distributions

I will mark this as homework, even though it's just general interest. The scenario is a little silly, because it's just a real world case for a theoretical problem I've been thinking about. A man ...
8
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1answer
159 views

Properties of bivariate standard normal and implied conditional probability in the Roy model

Sorry for the long title, but my problem is quite specific and hard to explain in one title. I am currently learning about the Roy Model (treatment effect analysis). There is one derivation step at ...
0
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0answers
37 views

Joint PDF change of variables

I now understand how to conduct a change of variables for a marginal PDF. Now, given two functions that define parameter's spatially: $C_A(x)$ and $C_B(x)$, is it possible to construct the Joint PDF, ...
1
vote
1answer
105 views

Is it normal to obtain better (smaller) P values in multivariate analysis compared to bivariate one?

If a multivariate design controls for other predictors when calculating the effect of a predictor, shouldn't it give paler P values (less significant ones, or less vivid odds ratios)? I am seeing ...
0
votes
1answer
162 views

Given known bivariate normal means and variances, update correlation estimate, $P(\rho)$, with new data?

I'm dealing with two correlated random variables which are modeled via a bivariate normal distribution. I have values for the means ($\mu_x, \mu_y$) and individual variances ($\sigma_x, \sigma_y$) of ...
0
votes
0answers
58 views

marginal of the bivariate normal wrt correlation

What is the distribution that results by marginalizing the correlation coefficient of the bivariate normal distribution, assuming a uniform prior in angular space: $$\int \; p(x,y|\mu,\Sigma(\theta)) ...
1
vote
1answer
91 views

Bivariate normal expectation of the sinus cardinal

I would like to get an analytical expression for $$\mathbb{E}\left(\frac{\sin(aX)}{aX}\frac{\sin(bY)}{bY}\right)$$ or at least an analytical approximation thereof, when $a,b$ are positive reals, and ...
1
vote
0answers
348 views

bivariate probit with endogenous covariate testing

I am interested in learning more about testing for the bivariate probit model with an endogenous treatment regressor. I have figured some stuff out -- summary below, since I don't see much on this ...
1
vote
1answer
444 views

Obtaining marginal distributions from the bivariate normal

Let $(X, Y)$ have a normal distribution with mean $(\mu_X, \mu_Y)$, variance $(\sigma_X^2, \sigma_Y^2)$ and correlation $\rho$. I want to know the corresponding marginal densities. All I found so far ...
1
vote
1answer
66 views

Finding an appropriate distance/divergence/similarity measure in a real 2D phase space

At first, I have to excuse my sloppy terminology, as I am pretty new to the whole topic. Imagine a real twodimensional phase space representing climate-related properties. I have a set of N variables ...
1
vote
2answers
480 views

Analysis and writing up results

I am doing research for my Masters studies and am battling a bit with the statistical side of my study. I used a survey and have captured all the data into SPSS. I have no statistical background, I ...
0
votes
1answer
242 views

KL divergence or similar “distance” metric between two multivariate distributions

I have a large dataset composed of many samples; each sample is as follows: imagine a grid indexed by i,j for a sample k, I have Y_k, where Y_k(i,j) is the probability density for k at (i,j) of ...
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0answers
843 views

Understanding the bivariate Poisson distribution

Researching bivariate Poisson over the web is no easy task unless you can make sense of the Greek symbols. I am familiar with Poisson and have a deep understanding of it. So could someone explain ...
3
votes
1answer
124 views

How to compute the distribution of a function of multiple random variables?

$X$ and $Y \sim U(0,1)$. Let $$\eqalign{ g(x,y) &= x &\text{ if } &x^2+y^2 \le 1 \\ &=2 &\text{ if } &x^2+y^2 \gt 1 }$$ and $Z = g(X,Y)$. How to find $F_Z(z), ...
1
vote
1answer
211 views

Farlie-Gumbel-Morgenstern Bivariate Gamma Distirbution

Given the variables $X$ and $Y$, which are correlated, $X\ge0$, $Y\ge0$ and each follow a gamma distribution with different shape parameters, i.e.,$X\sim Gamma(a_1,\alpha)$ and $Y\sim ...
5
votes
3answers
456 views

Probability of collision (two bivariate normal distributions)

I am trying to solve this problem on and off for the past couple of months but to no success. This was supposed to be a very small part of my PhD thesis in navigation but I guess I underestimated the ...
2
votes
1answer
787 views

What is a 'bagplot', or 'bivariate boxplot'?

I've found a paper which introduces the multidimensional (bivariate here) version of the boxplot - a bagplot. What is that bagplot exactly? I can see the series of nested polygons based on vertices, ...
2
votes
2answers
503 views

Correlation of bivariate grouped data?

Which test should I use, if I want to test the correlation between 2 bivariate grouped variables? The case is: I have asked several hotel owners about their feelings about the occupational rate of ...
5
votes
0answers
118 views

Joint distribution of two distances

Suppose there are three points in 3D space, each with coordinates $A_i=(X_i,Y_i,Z_i)\leadsto \mathcal{N}(\mu_i,\tau^2\mathbb{I}_3)$. We compute the distance between the three points, e.g. $D_{ij} = ...
4
votes
1answer
201 views

Inequality for bivariate normal distribution

Let $X_1$ and $X_2$ be bivariate normal with mean $\mu=(0,\mu_2)$, for any $\mu_2$, and correlation $\rho$. Consider the following inequality: \begin{align*} Pr\left\{|X_1| \ge ...
2
votes
1answer
232 views

When simulating a bivariate normal distribution, why is $\rho$ chosen instead of estimated from the data?

In a video lecture, MrProf shows the 3d-plot of a bivariate normal distribution $\mu_{x_1} = \mu_{x_2} = \sigma_1 = \sigma_2 = 1$ and chooses $\rho = 0.5$ . If stick to Mathworld, $\rho$ simply is ...
26
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2answers
2k views

Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian?

Somebody asked me this question in a job interview and I replied that their joint distribution is always Gaussian. I thought that I can always write a bivariate Gaussian with their means and variance ...
2
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0answers
117 views

Is it possible (or even usefull) to transform Log transformed data into Z-scores?

We have created a questionnaire. In this questionnaire there are different dimensions with different answering scales. Because of our rightly skewed data we log transformed our data. But here is the ...
1
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0answers
38 views

Approach to testing difference from a bivariate null distribution generated by randomization

I would like to test if each of the red observations is more extreme in variable xy[,2] than 95% of a null hypothesis (black dots) generated by randomization. I am ...
1
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2answers
526 views

Finding the Bayesian classifier for a bivariate Gaussian distribution

Very close to: Joint Gaussian of two Gaussians I am trying to find the Bayesian classifier for two classes given by the following bivariate Gaussian distributions: $$p(x|c_1) = N(\mu_1, ...
4
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1answer
546 views

Ellipse region shape from bivariate normal distributed data?

In my previous question I needed to help with ellipse region extraction and determine if point lies in that region or not. I ended up with this code: ...
5
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4answers
4k views

How to get ellipse region from bivariate normal distributed data?

I have data which looks like: I tried to apply normal distribution (kernel density estimation works better, but I don't need such great precision) on it and it works quite well. Density plot makes ...