Bivariate (two-variable) probability distributions.

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Estimation of stochastic parameters in bivariate Poisson model

I need to estimate parameters in bivariate Poisson model. Formulas for parameters: $\lambda_1= exp (\alpha_i-\beta_i+\delta)$ $\lambda_2= exp (\alpha_j-\beta_j)$ where delta is some constant, alpha ...
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joint probability distribution [closed]

Two four-sided dice with numbers 1,…,4 on the faces are thrown. Let the two discrete random variables U and V, be the sum of the outcomes for the individual dice and their product respectively. (i) ...
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Conditional expectation of bivariate normal

I have been reading Heckman (1979) and have tried to prove some result used (the paper points to a book which does not show the work either). I alter the notation a bit for clarity. Assume we have: $$\...
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Why use linear combinations of random variables?

I am in the process of learning bivariate distributions, especially bivariate normal disribution. I have noticed extensive use of linear combinations of random variables where the goal is to generate ...
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74 views

How to find $E|X+Y|^3$ from related information?

Assume that $$ E(X+Y)=E(X-Y)=0 $$ $$ V(X+Y)=3 $$ $$ V(X-Y)=1 $$ Show that $E|X+Y|\leq\sqrt3$. If in addition, it is given that $(X,Y)$ is bivariate normal, calculate $E|X+Y|^3$. For the 1st part, ...
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two wave panel probit with random effect = bi-variate probit

According to the paper "Estimating Dynamic Models of Quit Behavior" [Journal of Labor Economics], it said "Bivariate probit is equivalent to probit random effects if there are only two waves in a ...
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92 views

How to calculate the total probability inside a slice of a bivariate normal distribution in R?

I have a bivariate normal distribution composed of the univariate normal distributions $X_1$ and $X_2$ with $\rho \approx 0.3$. $$ \begin{pmatrix} X_1 \\ X_2 \end{pmatrix} \sim \mathcal{N} \left( \...
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1answer
51 views

Do I have a correlation here or not?

I am writing my thesis at the moment and I am pretty new with statistics. I tired to google around but I don’t know what to do anymore, since I don’t really find an answer in this case. I hope you can ...
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122 views

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 ...
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1answer
66 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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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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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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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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86 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} \int_{-\...
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1answer
110 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
60 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.
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53 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
40 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, $\mathrm{...
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132 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 ...
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136 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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59 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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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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63 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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76 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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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 ...
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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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28 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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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?
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127 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 $...
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266 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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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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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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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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36 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: $Y=X_1\...
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204 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 ...
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129 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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2k 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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81 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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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 ...
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108 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
238 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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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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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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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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116 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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301 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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105 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
141 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 ...