Questions tagged [bivariate]

Joint probability distribution of two variables.

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What is an example of a bivariate distribution that is dependent but nonlinear dependence?

I've seen plots of bivariate normal where the two variables are correlated. This results in a elongated peak area. What would a bivariate distribution look like if the two are dependent on each other ...
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Relation between bivariate survial function and cumulative density function

I am trying to understand why $Pr(T_1> t_1,T_2 > 2)=1-F_1(t_1)-F_2(t_2)+F_{12}(t_1,t_2)$ My derivation is as follows: \begin{align} Pr(T_1> t_1,T_2 > t_2) &=Pr(T_1> t_1\mid T_2>...
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23 views

bivariate normal distribution meaning [duplicate]

Does bivariate normal distribution mean the two random variables have normal distributions? is that enough for two random variables to have a bivariate normal distribution or are there some other ...
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22 views

Multiplying bivariate gaussians by a constant

Say I have the following : $$ (X, Y) \sim N_2(\mu, \Sigma) $$ Then what would be the distribution of $(2X,2Y)$ ? Let $\Sigma = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2\\ \rho\sigma_1\...
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27 views

How to find a such that X + aY is independent of X-aY for a bivariate distribution [closed]

Suppose $X$ and $Y$ are bivariate normal with equal variance, i.e. $[X, Y] \sim \mathcal{N} (0, \Sigma)$, where $$\Sigma=\begin{bmatrix}1&\rho\\\rho&1\end{bmatrix}$$ Find $a ≥ 0$ such that $...
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39 views

Show the bivariate normal cdf evaluated at (0,0) is increasing in the correlation coefficient

$\begin{bmatrix}\epsilon_{1}\\ \epsilon_{2}\end{bmatrix}\sim N(\begin{bmatrix}0\\0\end{bmatrix},\begin{bmatrix}1,\rho\\ \rho, 1\end{bmatrix})$. Show that the joint cdf evaluated at (0,0), i.e., $F_{\...
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46 views

Ellipse region shape from standard deviations

I need to draw a bivariate normal distribution ellipse based on this article. It says In the case of the bivariate normal distribution, both approximate and exact methods are available for ...
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21 views

Unbiased estimators of bivariate gaussian means

What are the best unbiased estimators of bivariate gaussian means given covariance matrix? Is there any such estimator that makes explicit use of the covariance matrix, and which is superior by any ...
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49 views

Formulas for higher order cumulants

I want to calculate higher-order joint cumulants for 2 variables. I calculated the higher order single-variable and bivariate moments numerically. Now I need to combine them into cumulants (upto the ...
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Karl Pearson correlation coefficient

How to prove that we can not measure the correlation coefficient between two variables using Karl Pearson correlation coefficient formula when there is a non linear relationship between them?
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Copula identification

I am looking a bivariate data and am trying to find a correct copula. I have been looking at the well known bivariate copulas (the archmedian copulas) but as I was looking, none of them seem to be the ...
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Modeling bivariate beta distributions in PyMC3

My goal is to perform a bayesian A/B test of probabilities of success in two groups considering a hypothesis about non-zero covariance between those probabilities. Bivariate beta distribution I am ...
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32 views

Not recovering true coefficient with recursive bivariate probit model on simulated data

I have built a simulated dataset to try to build my intuition about the recursive bivariate probit model. The challenge I'm running into is that I'm unable to recover the true coefficient in my ...
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1answer
30 views

Graphical interpretation of conditional density of a bivariate normal?

I am new to multivariate analysis and without a great math background I have been able to follow the book Applied multivariate statistical analysis up until this section... The way I am able to ...
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Fisher transformation

I am currently working with panel data, namely its correlations. The data is three-dimensional. I am looking for a way to aggregate the correlation coefficients. I found a model which uses the Fisher ...
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Computing the bivariate distribution from trivariate?

I appreciate in advance for any suggestions. Patton. A (2008), Modelling asymmetric exchange rate dependence, Int. Econ. Review.
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35 views

Subdivide Z into $X_1,X_2$, s.t. $Z=X_1+X_2$ and ($X_1,X_2$) obey bivariate normal

For a simulation in a research project, I am trying to randomly "appropriate" (meaning subdivide into two components) known values of $Z$ into $X_1$ and $X_2$ such that $Z=X_1+X_2$ in a way that ...
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How to compute the CDF of summation of two time series? (Using Copula)

My question is about the time series. For modeling the multivariate time series based on the conditional copula. I read two book “Copulas and their applications in water resources engineering”, and “...
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67 views

How can I find the distribution of the quantity $T(X,Y)$ given a sample from a bivariate normal distribution?

Let $(X_1,Y_1),\cdots,(X_n,Y_n)$ be a sample from a bivariate normal distribution with parameters $E(X_i)=\mu_1, E(Y_i)=\mu_2, Var(X_i)=Var(Y_i)=\sigma^2,$ and $Cov(X_i,Y_i)=\rho\sigma^2, i=1,\cdots,...
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Expected distance between (X, Y), where both X and Y are standrd normal random variabls and the origin

Let $(X, Y)$ be two independent standard random variable, with mean and SD being 0 and 1 respectively. What would be $E[\sqrt(X^2 + Y^2)]$, the expected distance between $(X, Y)$ and the origin.
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Bivariate exponential distribution $(S, T)$ with controllable correlation and $S\leq T$

I am trying to define a bivariate exponential distribution $(S, T)$ with marginals $S\sim\mathrm{Exp}(\lambda_S)$ and $T\sim\mathrm{Exp}(\lambda_T)$ for $\lambda_S > \lambda_T$. I would like the ...
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120 views

How can I find $\rho$, given $P(4 < Y < 16|X=5)=0.9544$?

Let $X$ and $Y$ have a bivariate normal distribution with $\mu_X=5, \mu_Y=10, \sigma^2_X=1, \sigma^2_Y=25, \rho >0$. If $P(4 < Y < 16|X=5)=0.9544$, I would like to find $\rho$. I know that ...
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How do I show that $Y=2\sqrt{X_1X_2}\sim$Gamma$(2p,1)$?

Suppose that $X_1\sim $Gamma$(p,1)$ and independently, $X_2\sim $Gamma$(p+1/2,1)$. Show that $Y=2\sqrt{X_1X_2}\sim$Gamma$(2p,1)$. This problem followed a section on bivariate transformations, so I ...
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Example: Writing the joint PDF $f(x, y)$ as the product of a marginal and a conditional probability function

I am presented with the following notes on Bivariate distribtions: If we can write the joint probability density function $f(x, y)$ of a pair of random variables $(X, Y)$ as the product of a ...
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Sum of Two Bivariate Normal Distributions

I'm just confused on how to set up and start this problem. I'm confident that once I start down the right path, I'll have little issue. Let $p_1$ denote a bivariate normal distribution $N(0, 0, 1, 1, ...
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Bivariate Dist Study Question Help - determine joint PMF and P( … )

I am in a prob. models class. Current module is on Bivariate and Multivariate Distributions. The question below has me stumped though. It is from a study guide and I would like to know the answer ...
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31 views

Question regarding variances in Simple Linear Regression [closed]

In a linear bivariate model, why is Var(u|x) = Var(y|x)?
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50 views

question about bivariate gaussian [duplicate]

I have a bivariate normal distribution $$f(x,y)=\frac{1}{2\pi \sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}\exp\left(-\frac{z}{2(1-\rho^{2})}\right) $$ where $$z=\frac{x^2}{\sigma_{1}^2}+\frac{y^2}{\sigma_{...
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Correlation between two variables.. but does it make sense?

I'm a grad student and I'm struggling with my first paper. I found a correlation between two variables and it turns out to be statistically significant. The variables are log-transformed. The data ...
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Bivariate Data Variables

When comparing two different variables to determine whether or not one determines the other, do both data sets of the variables need to be from the same year? Such as life expectancy from 2004 and HDI ...
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1answer
96 views

Variance of marginals of truncated bivariate normal distribution

I have a truncated bivariate normal distribution $$f(x,y)=\begin{cases}\frac{1}{2\pi \sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}\exp\left(-\frac{z}{2(1-\rho^{2})}\right) &, |x| \leq a\\0 &, |x| &...
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Integration of (Phi(x)-Phi(y))^2d(F(x, y)

How to integrate the following? Thanks
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1answer
26 views

Two simultaneous time series vars. Want “confidence” intervals for one's prediction (not forecast per se) of the other

Let me know if this is a duplicate. This seems to differ from other time series questions I've seen in that I'm not trying to forecast the future. Maybe that will mean only a difference in the ...
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1answer
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Lcross Confidence Envelopes - R software

I have performed an Lcross examination in R with the following code: ...
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47 views

correlation coefficient bivariate normally distributed [duplicate]

Suppose that X,Y and X,Z are bivariate normally distributed. We have $E(X)=0, Var(X)=10$, $E(Y)=0, Var(Y)=6$ and $ρ_{xy}=0.87$ Moreover, $E(X)=0, Var(X)=10$, $E(Z)=0, Var(Z)=4$ and $ρ_{xz}=0.87$ ...
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Simplification of bivariate normal $\phi_2(x,y,\rho)$ at $y=y_F$ (i.e. fixing one of the axes)

Suppose we start off with the traditional standard bivariate normal distribution: $$\phi_2(x,y|\rho,\mu_x=0,\mu_y=0,\sigma_x=1,\sigma_y=1)=\frac{1}{2\pi\sqrt{1-\rho^2}}\exp \left(-\frac{x^2-2\rho x y ...
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Probailty of Ygreater than or equal to X^2

I want to solve the below mentioned question which involve transformation or change of variable technique
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Do outliers affect bivariate correlation? [closed]

I recently found a significant positive correlation for three variables (anxiety, depression, and Fear of Missing Out). FOMO is my variable of interest. I did a one sample t-test to measure the ...
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57 views

What is the variance of the difference of two means?

I'm trying to express $\mathrm{Var}(\mu_x - \mu_y)$ in terms of $\rho$, $\sigma_x$ and $\sigma_y$, where $\mu$ denotes the mean of the random variable. Firstly:\begin{align*} \mathrm{Var}\left(\sum_{...
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Question Regarding Zero Conditional Mean

Hi I am a beginner to econometrics! I have been dealing with bivariate regression. We use the formula $y = \beta_0 + \beta_1 x$. I am told that if $E(u\mid x) \ne 0$ then the estimate of the slope ...
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832 views

How to estimate a bivariate probit (biprobit) model in R with a different set of explanatory variables? [closed]

I'm trying to estimate a bivariate probit model (also called biprobit model) in R where the set of explanatory variables is different for both binary outcomes. Thus, my setting is: \begin{align} Y_1^*...
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2answers
182 views

Bivariate vs. multivariate analysis

I have been reading a few papers lately that has done both bivariate and multivariate analysis on their data. What I have seen most of the times is that they usually do the bivariate analysis first, ...
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464 views

MLE of Parameters of Bivariate Normal Distribution

I am working through find the maximum likelihood estimators of the bivariate normal distribution, without using matrices. I have the following density function: $f(Y_1,Y_2) = \frac{1}{2\pi\sigma_1\...
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Variance of bivariate normal distribution plus normal distribution

Problem: $W = -27 + 0.3X + 0.45Y + E$ The pair $\begin{bmatrix} X \\ Y \end{bmatrix}$ behaves like a bivariate normal with vector of averages $\begin{bmatrix} 156 \\ 86 \end{bmatrix}$ and ...
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126 views

Unexpected behavior in inhomogeneous Cross K Function (Kcross.inhom) [closed]

I am currently analyzing a point pattern in R using the "spatstat" package. I am comparing two different areas, therefore I made two plots for each area (first two plot-left-area1; second two plots-...
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501 views

I want to create a random sample of length n from a normal multivariate distribution

I have a problem in understanding this question, especially this part: "generate a random sample of length n from a normal multivariate" This is what I have done using the R package mvtnorm: ...
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195 views

How to justify that $(Y_1,Y_2)$ is not bivariate normal without finding its exact distribution?

Suppose $X_1$ and $X_2$ are independent $N(0,1)$ variables. Define $$Y_1=X_1\,\text{sign}(X_2)\quad,\quad Y_2=X_2\,\text{sign}(X_1)$$ I have to show that $(Y_1,Y_2)$ is not bivariate normal ...
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Knowing correlation, bivariate normality, means and standard deviations, find values probability

After months of study I still do not get it. I apologize (see: Estimating values knowing their Pearson's r and their means and standard deviations) Imagine, for example, I have two bivariate ...
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91 views

Solving a double integral over transformations of joint bivariate standard normal values

My problem is about calculating the covariance between transformations of two test statistics based on the correlation between these test statistics. Let $X$ and $Y$ be two test statistics whose joint ...
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229 views

Simulate Bivariate Beta Distribution , BIBETA(6, 20, 2) in R

I need to simulate bivariate beta distribution, $BIBETA(6, 20, 2)$ in r. I am looking for a package/ code that would generate bivariate beta distribution. I couldn'...

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