Bivariate (two-variable) probability distributions.

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Find data and confidence “ellipses” (regions?) for a bivariate median?

I'm wondering about ways to compute data and confidence ellipses around a bivariate median. For example, I can easily compute a data ellipse or a confidence ellipse for the bivariate mean of the ...
1
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1answer
47 views

Bivariate probit model with sample selection

Could you please provide an example and explanation why to use the bivariate probit model with sample selection? In this context, to what sample selection bias refers to?
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0answers
25 views

Finding a valid bivariate distribution

For each of the following functions, determine the constant $c$ so that $f(x,y)$ satisfies the conditions of being a joint pmf for two discrete random variables $X$ and $Y$: (c) $f(x,y) = c$; $x$ and ...
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0answers
30 views

Bivariate Normal Distribution [duplicate]

Suppose that $(X,Y)$ has the probability density function given below: $f(x,y)=\frac{1}{\sqrt{3}{\pi}} e^{-\frac{2}3(x^2-xy + y^2)}, (x,y)\in \mathbb{R}^2$ a) I want to find the density function of ...
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0answers
11 views

Bivariate Normal Distribution of a Random Variable (RV) and the sum of that RV with another [duplicate]

Let X and E be two independent normal random variables. Let Z=X+E. I've read that X and Z would have a joint (bivariate) Normal distribution. Can someone tell me why? Further, under what conditions ...
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0answers
49 views

How to calculate integral of density under constraint?

Given a bivariate standard-normally distributed random variable $Y=[Y_1,Y_2]^T$ with density $\phi(Y_i,Y_2)$, the probability of $Y_1>0$ is simply $$P(Y_1 > 0)=1- \int_{-\infty}^{\infty} ...
2
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1answer
92 views

How to construct a bivariate distribution from marginal distributions with a predefined correlation

I would like to generate zipf and lognormal random variables with a particular correlation. Then, I would like to find their bivariate distribution. What approach should I follow?.
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1answer
36 views

Finding the Coefficients of regressors

I understand how to find the coefficients of a bivariate regression and univariate regression w/o an intercept, i.e: Univariate: Y = BX + e ...
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1answer
46 views

What is the “systematic component of the model”, in bivariate linear regression?

I need to discuss the systematic component of the model in bivariate linear regression, but what is it in the first place? I have never come across this terminology in our class textbooks.
3
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1answer
44 views

From the bivariate poisson to the Skellam (or Poisson Difference) distribution

I am looking at how to calculate the Skellam distribution (https://en.wikipedia.org/wiki/Skellam_distribution) from the bivariate poisson distribution. I understand that the Skellam comes from ...
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1answer
33 views

Difference between $\mathrm{Poisson}(x_1)$, $\mathrm{Poisson}(x_2)$ and $\mathrm{BPoisson}(x_1, x_2)$

I am trying to find out the difference between treating two random variables as poisson distributions, $\mathrm{Poisson}(x_1)$, $\mathrm{Poisson}(x_2)$, and using a bivariate poisson, ...
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0answers
83 views

Multivariate asymmetric generalized gaussian distribution

I would like to write the distribution of a multivariate asymmetric generalized gaussian distribution and plot the result with Matlab. So far I was able to write the code to create a bivariate ...
8
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2answers
111 views

Is joint normality a necessary condition for the sum of normal random variables to be normal?

In comments following this answer of mine to a related question, Users ssdecontrol and Glen_b asked whether joint normality of $X$ and $Y$ is necessary for asserting the normality of the sum $X+Y$? ...
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1answer
25 views

Two dimensional PDF candidate for barbell like distribution

I have this empirical sample, which I drew on the scatter plot. It's got to be unimodal according to theory, and it surely does look like one. A simple way to model it is with bivariate normal. ...
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1answer
52 views

What would be Maximum Likelihood Function of an Independent Bivariate Normal Sample and how it works?

What is the maximum likelihood function of an independent Bivariate normal sample $(x_i, y_i)$, where the mean is known as a vector of $(\mu_x, \mu_y)$, and the variance is known to be some sort of a ...
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0answers
19 views

Contradicting results of a multivariate and pairwise/bivariate Johansens reduced rank test

I was performing Johansen's reduced rank test on a group of 3 commodity price series $X$, $Y$, and $Z$ ($n=3$; lag-length selection based on the SBIC but at least 2; unrestricted constant model (case ...
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0answers
54 views

Is it possible to use F-statistics to test if beta=1 and alpha=0 in the CAPM?

Is it possible to use F-statistics to test if beta=1 and alpha=0 in the CAPM even though the joint distribution of excess return of equity and excess return of the market are not bivariate normal? ...
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1answer
71 views

Normality of conditional pdf(s) does not imply BVN

Show by means of example that the normality of conditional pdf(s) does not imply that the bivariate density is normal. I know of the following example that if $\ U,V,W $ and $\ T$ are independently ...
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20 views

Deriving the Bivariate Normal Distribution from Normal Distributions [duplicate]

If $X \sim N(0,{a^2})$, $Y \sim N(0,{b^2})$ and $Corr(X,Y) = \rho $, then can we say that $(X,Y) \sim BVN(0,0,{a^2},{b^2};\rho )$? If this is true, then can someone please tell me how can I ...
0
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1answer
13 views

Variance of bivariate values

If I have two lines of regression ($y$ on $x$ and $x$ on $y$) and I know $\sigma(x)$, why isn't the variance of $y$ equal to $m^2 \sigma(x)^2$ where $m$ is the slope?
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1answer
26 views

Bivariate Normal Distribution Probability Calculations

While doing questions related to Bi-variate Normal Distribution (BVN) I came across the following question which I was unable to decide how to proceed with: If $(X,Y) \sim BVN(0,0,1,1;\rho )$ then ...
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2answers
91 views

Bivariate normal distribution with $|\rho|=1$

I have deduced the bivariate normal density function. However am unaware of what happens when the correlation coefficient $\rho$ tends to 1 and -1?
0
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1answer
93 views

Generate bivariate random numbers from joint distribution function

I have an empirical joint distribution function $ \hat{F}(x_1,x_2) = Pr(X_1 < x_1, X_2 < x_2) $ Can I generate bivariate random number from this distribution with a certain condition such as ...
3
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2answers
179 views

Dependent identically distributed random variables

If $X_1$ and $X_2$ are dependent identically distributed, can we show that $Pr(X_1>X_2) = Pr(X_2>X_1)$? For i.i.d, it is obvious, but what if they are dependent?
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40 views

Can I use Multiple R of Multiple Linear regression to explain a relationship between independent and dependent variables

In my questionnaire, I have four independent variables (under each independent variable 3 - 6 questions are given) which will be measured by the answers of the questions given under each variable. ...
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16 views

Bivariate Analysis - P Charts

Are P-charts considered to be bivariate analysis, since it's examining 2 variables, defects and sample size?
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18 views

Conditional Expectation Bivariate Normal with OLS

I am working on the effects of omitting variables in a regression for my thesis. I already found a lot about the bias that is created by omitting variables, but not so much about what this does for ...
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0answers
34 views

Conditional expectation of error based on multivariate normal variables

I have the following situation; I know the "true" model behind my regression but I am intentionally omitting some variables/regressors to simplify the problem. Suppose the true model is: ...
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2answers
154 views

Probability distribution of the magnitude of a circular bivariate random variable?

I'm very new to this topic. I have a distribution similar to the picture below but with the center at zero. As I said, I'm very new to this, but if I understand correctly, if there was no hole in ...
0
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1answer
97 views

Bivariate K function for inhomogeneous spatial point processes

I have some inhomogeneous spatial point patterns of individuals in a cactus population. I also have marks, such as "diseased"x "healthy" individuals, and "adult" x "juvenile". I've already computed ...
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1answer
1k views

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 ...
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1answer
80 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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60 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 ...
0
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1answer
31 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 ...
3
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1answer
100 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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1answer
213 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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0answers
60 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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82 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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23 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
113 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 ...
10
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1answer
295 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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1answer
101 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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1answer
118 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 ...
0
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1answer
152 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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0answers
167 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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3answers
201 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
455 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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0answers
54 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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1answer
127 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 ...
8
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2answers
189 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 ...