Bivariate (two-variable) probability distributions.

learn more… | top users | synonyms

1
vote
1answer
39 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
votes
1answer
34 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 ...
1
vote
1answer
30 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, ...
1
vote
0answers
45 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
votes
2answers
93 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 $X+Y$? ...
1
vote
1answer
23 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. ...
1
vote
1answer
37 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 ...
0
votes
0answers
17 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 ...
0
votes
0answers
43 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? ...
4
votes
1answer
68 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 ...
1
vote
0answers
19 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
votes
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?
1
vote
1answer
24 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 ...
2
votes
2answers
79 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
votes
1answer
61 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 ...
2
votes
2answers
135 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?
0
votes
0answers
36 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. ...
0
votes
0answers
14 views

Bivariate Analysis - P Charts

Are P-charts considered to be bivariate analysis, since it's examining 2 variables, defects and sample size?
1
vote
0answers
16 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 ...
0
votes
0answers
32 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: ...
2
votes
2answers
103 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
votes
1answer
61 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 ...
1
vote
1answer
690 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 ...
2
votes
1answer
75 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. ...
0
votes
0answers
54 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
votes
1answer
30 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
votes
1answer
95 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 ...
1
vote
1answer
181 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 ...
2
votes
0answers
57 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 ...
0
votes
0answers
41 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 ...
0
votes
0answers
74 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 ...
1
vote
0answers
22 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 ...
1
vote
1answer
107 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
votes
1answer
282 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= ...
0
votes
0answers
16 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 ...
2
votes
1answer
84 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 ...
1
vote
1answer
87 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
votes
1answer
134 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 ...
2
votes
0answers
160 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 ...
6
votes
3answers
195 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 ...
2
votes
1answer
349 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 = ...
0
votes
0answers
53 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. ...
1
vote
1answer
110 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
votes
2answers
179 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 ...
1
vote
1answer
71 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 $N(0,0,1,1,\rho)$ ...
4
votes
1answer
2k 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
votes
1answer
39 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 ...
8
votes
2answers
2k 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 ...
0
votes
1answer
265 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 ...
3
votes
0answers
91 views

Is it statistically correct to compare two Spearman's $\rho$-values?

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 ...