Is it possible for two random variables to be negatively correlated, but both be positively correlated with a third r.v.?

Would it be possible for two variables to be negatively correlated with one another, yet be positively correlated with a third variable? Are there any concrete examples?

Answer

Certainly. Consider multivariate normally distributed data with a covariance matrix of the form

(1+1+++1).

As an example, we can generate 1000 such observations with covariance matrix

(10.50.50.510.50.50.51)

in R as follows:

library(mixtools)
set.seed(1)
xx <- rmvnorm(1e3,mu=rep(0,3),
    sigma=rbind(c(1,-.5,.5),c(-.5,1,.5),c(.5,.5,1)))
cor(xx[,c(1,2)])
cor(xx[,c(1,3)])
cor(xx[,c(2,3)])

The first two columns are negatively correlated (ρ=0.5), the first and the third and the second and the third are positively correlated (ρ=0.5).

Attribution
Source : Link , Question Author : aquaplane , Answer Author : Stephan Kolassa

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