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3 views

Example of two *correlated* normal variables whose sum is not normal

I am aware of some nice examples of pairs of correlated random variables which are marginally normal but not jointly normal. See this answer by Dilip Sarwate, and this one by Cardinal. I am also ...
0
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1answer
17 views

Finding $p(\tilde{y}|x)$ given measurement model and error distribution

Given two measurements of a variable $x$: $\tilde{y_1}=x+e_1$ $\tilde{y_2}=x+e_2$ where $e_1,e_2$ are zero-mean random variables following a bivariate normal distribution, with a known joint ...
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2answers
300 views

What is the maximum likelihood estimate of the covariance of bivariate normal data when mean and variance are known?

Suppose we have a random sample from a bivariate normal distribution which has zeroes as means and ones as variances, so the only unknown parameter is the covariance. What is the MLE of the ...
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0answers
39 views

Bivariate Poisson regression: Error: “in `contrasts<-`(…): contrasts can be applied only to factors with 2 or more levels ”

I realize this error message has been posted before, but the solutions previously provided does not work for me, so here goes: I am working with a data.frame, which looks like this: ...
0
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1answer
61 views

Generate data from a bivariate power-law distribution in R

I need to generate data from a random vector that follows a bivariate power-law: $$ f_{X,Y}(x,y) = \frac{C}{XY} \left(\frac{X}{X_0} \right)^{-\alpha} \left(\frac{Y}{Y_0} \right)^{-\beta} , $$ where ...
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0answers
43 views

Statistically correct to compare two Spearman's $\rho$-Vaules?

I have different variables (A, B, C, D) which are all ordinal scaled. I have another variable (Z) which is ordinal scaled, too. My sample size is approx. 1.500. Is it in the statistical sense ...
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0answers
51 views

If $X, Y$ are jointly standard normal with correlation $r$, and $a, b$ are constants, what is $p(Y < b | X < a)$?

The application here is interpreting the correlation coefficient $r$ in terms of $X$'s ability to predict $Y$ for extreme values of $X$. For example, if $r = .8$, then what is $p(Y < 0 | X < ...
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1answer
43 views

Statistics Test for bivariate frequency table

First time user here! I have some troubles in finding some statistic method that would fit my purpose for testing bivariate frequency table. Below is a sample frequency table I have created with ...
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0answers
28 views

On tests for correlation in case of bivariate observations

We have bivariate observations {($X_i, Y_i$), i=1,2,...n}. We are interested in testing if they are uncorrelated. One test is based on $r$ (correlation coefficient): $r\sqrt{\frac{n-2}{1-r^2}}\sim ...
2
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3answers
181 views

In SPSS, how to compare two scatterplots of separate bivariate data to determine if the distributions are similar or not similar?

In SPSS, I have created scatterplots of two continuous variables (X & Y) for two separate groups (G & P). Visual inspection comparing the scatterplots from the two groups suggests that ...
6
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1answer
191 views

Why can't one generalize the Kolmogorov-Smirnov test to 2 or more dimensions?

The question says it all. I've read both that one can't generalize KS to a dimension equal or larger than two, and that famous implementations like that in Numerical Recipes are simply wrong. Could ...
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0answers
56 views

How to derive the characteristic function of a polar coordinates representation of a bivariate normal

Suppose to have a bivariate normal variable $\mathbf{x}=(x_1,x_2)$ with mean $\mu$ and covariance matrix $\Sigma$. I move from $\mathbf{x}$ to $(\theta,r)$ where $x_1 = r \cos \theta$ and $x_2 = r ...
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0answers
39 views

Sample size required for bivariate normal distribution

I am new here. Would like to ask a question on the sample size requirement for hypothesis testing following. If I am drawing samples with 2 non correlated variables (say x,y) from a bivariate normal ...
2
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1answer
77 views

Distribution of distance from center of sample group

We have a bivariate normal process where $X, Y \sim N(0, \sigma)$, with no covariance. (For convenience we can assert that $\sigma = 1$, or that we have a good estimate for its value.) What is the ...
2
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1answer
96 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
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2answers
68 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 ...
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0answers
14 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
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1answer
29 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 ...
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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
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0answers
29 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 ...
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2answers
141 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
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1answer
41 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
14 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} ...
3
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1answer
61 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 $X \leq \mu_X + k\sigma_X$ ...
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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
26 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
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3answers
253 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 ...
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2answers
197 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 ...
1
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1answer
256 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
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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 ...
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1answer
81 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
75 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
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1answer
84 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
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1answer
93 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
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1answer
50 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
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1answer
109 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
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1answer
303 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
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2answers
123 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
165 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
50 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, ...
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1answer
156 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
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1answer
249 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 ...
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1answer
110 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 ...
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0answers
442 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 ...
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1answer
754 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 ...
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1answer
71 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
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1answer
559 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
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1answer
284 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
1k 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
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1answer
134 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), ...