Correlation between standard normal and T-distributed random variables

Question

Suppose you have two random variables, one uniformly distributed on interval [-3, 3] and the other standard normal. Is it possible for them to have correlation 1 or -1? What if the first variable is T-distributed?

Answer 1

Not possible in either case. If the correlation coefficient of $X,Y$ is $\pm 1$ then $X-EX=c(Y-EY)$ for some constant $c$. This is the condition for equality in Holder's/ Cauchy-Schwarz inequality. With the given distributions this is not possible.

Answer 2

They are not the same distribution or a linear transformation of each other so the correlation cannot be $+1$ or $-1$.

It may be possible to find the bounds on the correlation analytically, but here is an easy approximation to the maximum possible correlation using R, where $X_{N(0,1)}=\Phi^{-1}\left(\frac{X_{U[-3,3]}+3}{6}\right)$:

p <- ppoints(10^7)
Xu <- qunif(p,-3,3)
Xn <- qnorm(p)
cor(Xu,Xn)
# 0.9772051

For a Student $t$-distribution, you cannot find a correlation with $\nu=1$ or $2$ degrees of freedom as the distribution does not have a finite variance.

For $\nu=3$ this approximation method is not accurate beyond the second decimal place, but you can see that the maximum correlation is smaller, due to the heavy tails of this $t$-distribution.

Xt3 <- qt(p,3)
cor(Xt3,Xn)
# 0.911941

For large $\nu$ the maximum correlation with a normal distribution is high and increases towards $1$ as $\nu$ increases. For example with $\nu=30$:

Xt30 <- qt(p,30) 
cor(Xt30,Xn)
# 0.9997653

Here are qqnorm plots (using fewer points) for the three examples. The first two are visually non-linear, while the third is close

• for the uniform distribution:

• for the $t_3$ distribution:

• for the $t_{30}$ distribution: