Standard deviation of a function of random variables

Question

I was a bit confused recently with why two jointly distributed variables will exhibit a smaller scatter as the correlation between the random variables increases (assuming only positive values for the correlation coefficient), but the standard deviation of the function of the sum or difference between these two variables would increase as the correlation between the random variables increases (see example., for the latter).

So I wrote a small code a snippet of which is below:

    import numpy as np
    from scipy import stats

    from statsmodels.distributions.copula.api import (
        CopulaDistribution, IndependenceCopula, GaussianCopula)
    import random

    random.seed(10)
    number_sample = 100000
    
    marginals = [stats.norm, stats.norm]
    
    # Case 1: independence
    # JOINT PDF
    joint_dist = CopulaDistribution(copula=IndependenceCopula(), marginals=marginals)
    sample1 = joint_dist.rvs(number_sample, random_state=10)
    #PDF X - Y
    sample_xy1 = sample1[:, 0] - sample1[:, 1]
    print(np.std(sample_xy1))
    
    # Case 2: Dependence
    # JOINT PDF
    copula = GaussianCopula(corr=[[1,0.5],[0.5,1]], k_dim=2, allow_singular=False)
    joint_dist = CopulaDistribution(copula, marginals)
    sample2 = joint_dist.rvs(number_sample, random_state=10)
    
    #PDF X - Y
    sample_xy2 = sample2[:, 0] - sample2[:, 1]
    print(np.std(sample_xy2))

The result is that the stdev of the function $Z=(X-Y)$ when $X$ and $Y$ are independent equals $(X^2+Y^2)^{0.5}$, but when they are correlated the value is smaller, although according to theory it should be larger.... What am I doing wrong?

Answer 1

let's put in the covariance term so that we can see how correlation impact the variance of 𝑋±𝑌: Var(𝑋±𝑌)=Var(𝑋)+Var(𝑌)±2Cov(𝑋,𝑌)

In your case, it should be: Var(𝑋-𝑌)=Var(𝑋)+Var(𝑌)-2Cov(𝑋,𝑌) where you assumed X positively correlates with Y which means the covariance term is positive.

So the variance/stdev of Z should be theoretically smaller.