How to Construct Correlated Normal Variables from Independent Normals?

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SUMMARY

This discussion focuses on constructing correlated normal variables Y1 and Y2 from independent standard normal variables X1 and X2. The relationships established are Y1 = s1X1 + m1 and Y2 = bX1 + cX2 + m2, where b and c are defined by the correlation coefficient ρ and the variances σ1 and σ2. The method relies on linear combinations to achieve the desired correlation, with the condition b² + c² = σ2². The approach is validated through the experience of the contributor, mathman, who emphasizes the simplicity of adding means and the flexibility of setting one coefficient to zero.

PREREQUISITES
  • Understanding of standard normal distribution and properties
  • Familiarity with linear combinations of random variables
  • Knowledge of correlation coefficients and their implications
  • Basic statistical concepts such as mean and variance
NEXT STEPS
  • Explore the derivation of linear combinations in multivariate normal distributions
  • Learn about covariance matrices and their role in constructing correlated variables
  • Study the implications of setting coefficients to zero in statistical models
  • Investigate the application of correlated normal variables in statistical simulations
USEFUL FOR

Statisticians, data scientists, and researchers involved in probabilistic modeling and simulation who need to understand the construction of correlated normal variables from independent sources.

gradnu
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I have two independent standard normal random variables X1,X2. Now I want to construct two new normal random variables Y1,Y2 with mean\mu1, \mu2 and variance (\sigma1)^2, (\sigma2)^2 and correlation \rho.
How do I approach this problem?
 
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Y1=s1X1+m1
Y2=bX1+cX2+m2
where b2+c2=s22
b=rs2, therefore c=s2(1-r2)1/2
 
Thanks mathman.
But what was your thought process? How did you come up with these relations?
 
gradnu said:
Thanks mathman.
But what was your thought process? How did you come up with these relations?

From long past experience I know that to get correlated normal variables from uncorrrelated standard normal, you just need a linear combination. Adding the desired means is obvious. Also since there are four free coefficients and there are only three conditions, I just set one coefficient to 0.
 

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