Bivariate (two-variable) probability distributions.

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Seemingly unrelated bivariate probit for endogeneity: interpretation of Rho

I would like to estimate the effect of health insurance coverage on type of healthcare provider chosen--either public or private--at last illness using a nationally representative sample of people in ...
2
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1answer
59 views

Copula density function

In the equation $h_t(x,y|...) = ...$, can anyone explain me why the first derivatives of the marginal distributions are included? $H_t$ is a distribution function and $h_t$ its density function. ...
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32 views

2D Kernel density estimation with uncertainties

I would like to perform bivariate KDE with Gaussian kernels (preferably using Python, or R) of a dataset with heteroscedastic uncertainties. What would be the correct way to do this: to rescale a ...
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25 views

What is this bivariate distribution called and how to make it posterior?

I am trying to make this bivariate density function as posterior f(x,y) = k x^2 exp( - x y^2 - y^2 + 2y - 4x) and try jags instead of implementing in R as in ...
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53 views

Bivariate distribution: beta and binomial

Consider a pair of RVs $X$ and $Y$, with the following conditional distributions: $$X | Y=y \sim Binom(L, y)$$ $$Y | X=x \sim Beta(\alpha + x, \nu)$$ where $L$, $\alpha$, and $\nu$; are all ...
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78 views

Generating random variables from copula function at a given joint probability?

I have one copula function, let's say a 2 dimensional Normal Copula with parameter of 0.5. I want to generate random variable pairs at given copula probability (e.g. 0.9). How can I do that? (since ...
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37 views

Proving a “well-known” result regarding the distribution of a normally distributed random variable

In an important project work, I would like to include a "proof" of the following, but have unfortunately been unable to readily compute it myself. I am aware that this is a flaw on my part, but ...
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15 views

Self-study (Expectation Maximization on Bivariate Normal Distribution)

I see this example is also "classic", and I am attempting to understand how to approach it. I have an iid sample drawn from a bivariate normal distribution with mean vector ($\mu_1, \mu_2$) and ...
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38 views

Bivariate ARIMA model in Stata

I am having troubles fitting a bivariate ARIMA model in STATA. Is there such a capability at all? I can choose dependent and independent variables but once I set them to be mortality and alcohol ...
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17 views

density function of bivariate normal with almost singular correlation matrix [closed]

Let $X$ be a bivariate normal distribution with mean $[0,0]^T$ and covariance matrix \begin{pmatrix} 1&\rho\\ \rho&1 \end{pmatrix} with $\rho<1.$ I am looking for the behaviour of the ...
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1answer
88 views

Fit data to a bivariate function

I want to fit my (x,y,z) data points to a function. You can see the data on Fig.1. The data is symmetric along the main diagonal. To understand my data I have studied (y,z) curves at different ...
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239 views

calculation threshold for minimum risk classifier?

Suppose Two Class $C_1$ and $C_2$ has an attribute $x$ and has distribution $ \cal{N} (0, 0.5)$ and $ \cal{N} (1, 0.5)$. if we have equal prior $P(C_1)=P(C_2)=0.5$ for following cost matrix: $L= ...
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15 views

Inference in bivariate continuous distributions

We have two nodes in different positions, which are represented by two random variables X,Y, with two prior bivariate continuous distributions, p_X , p_Y. f(X,Y,U,V) is a constraint on both ...
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1answer
64 views

Derive the Gibbs sampler for this bivariate distribution

I understand the theory of Gibbs sampling. It is an iterative sampling algorithm that defines, sequence of random variables with the property of a Markov chain. Specifically, I choose any starting ...
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37 views

Drawing confidence regions around a multimodal bivariate kernel density plot

The following plot gives a bivariate kernel density plot (made with kde2d in MASS) of a deterministic functional evaluated at ...
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1answer
47 views

bivariate normal distribution probability

We have two genes X and Y. Let $(X,Y)\sim N(\mu_x=9,\mu_y=10,\sigma^2_x=3,\sigma^2_y=5,\rho\sigma_x\sigma_y=2)$. To find $P(X+0.5<Y)$ the probability that the sample mean for the second gene ...
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124 views

What's the Mode of a Bivariate Poisson Distribution?

I have been looking at the bivariate Poisson distribution and I am wondering if there is close form expression for the mode of this distribution. I know the mode of the univariate Poisson distribution ...
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143 views

Define the joint pmf of a particle moving randomly on a grid

A particle starts at (0,0) and moves in one-unit independant steps with equal probabilities of $\frac{1}{4}$ in each of four directions: north, south, east, and west. Let $S$ equal the east-west ...
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1answer
128 views

Find the pmf of a bivariate distribution when rolling a black and red four-sided die

Roll a pair of four-sided dice, one red and one black. Let $X$ equal the outcome on the red die and let $Y$ equal the sum of the two dice. Define the joint pmf on the space. So far I have $X = ...
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47 views

Observed vs expected values for a dataset with repeated measures

I am hoping you all may be able to help me select the appropriate statistical analysis for a subset of my data. I have a population of monkeys that consists of 50 individuals in 3 groups. ...
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102 views

How to fit Gumbel Copula

I am trying to apply Gumbel Copula in R by using "copula" pkg. The parameter "alpha" of gumbel copula is 1.016. The copula structure is: gum.cop=archmCopula(family="gumbel",dim=2,param=alpha) But the ...
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1answer
81 views

Heckman 2-step Error Assumption

first question on StackExchange; thank you for having me. I am trying to really nail the intuition for the Heckman sample selection model. One little thing that is bothering me is the assumption ...
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127 views

Limits on conditional expectation with normal margins and specified (Pearson) correlation

I saw the following question on another forum: "Suppose that both height and weight of adult men can be described with normal models, and that the correlation between these variables is 0.65. If a ...
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30 views

Mix of two bivariate distributions (two correlations hidden in data)

We have two metric (continuous) variables, say $X$ and $Y$ and are interested in a correlation between $X$ an $Y$. Actually, a correlation test (Pearson or Spearman) is not significant, i.e. it does ...
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67 views

Bivariate normal density function

Let $F(\cdot,\cdot, \mu_1, \mu_2,\sigma_1^2,\sigma_2^2,\rho)$ denote the d.f. $(X,Y)$. Show that $$\Bigg(\frac{X-\mu_1}{\sigma_1}, \frac{Y - \mu_2}{\sigma_2}\Bigg)$$ has a $N(0,0,1,1,\rho)$ ...
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851 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 ...
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36 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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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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73 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: ...
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153 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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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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68 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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82 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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33 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 ...
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720 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 ...
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244 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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73 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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79 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 ...
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1answer
89 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 ...
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1answer
202 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 ...
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94 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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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 ...
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41 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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58 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 ...
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31 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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191 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 ...
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1answer
48 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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112 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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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 ...
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37 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 ...