I have two variables obtained from a tension test: modulus of elasticity (moe) and peak tensile strain.
The values for moe are as follows:
moe = [1575.3 1277.4 3667.1 5589.9 4079.7 1449.7 6654.5 1938.9 993.7 2178.3 1505.7 1727 12570.9 2735.2 2272.3 3168.6 2454 15470.3].
The values for peak strain are:
strain = [0.013919 0.019329 0.006024 0.003251 0.009376 0.016726 0.003944 0.010426 0.019642 0.015626 0.01508 0.017184 0.001958 0.013546 0.014779 0.010526 0.007719 0.000999].
Based on the data, it appears that the lognormal distribution is the best fit for moe, while for peak strain, an empirical cumulative distribution function (CDF) was used as the lognormal distribution did not provide a good fit. All the values are positive.
The correlation coefficient between moe and peak strain was found to be approximately -0.79. Now, I'm facing difficulties in generating correlated random variables (say a 1000 of them) for moe and strain with the same correlation coefficient (-0.79), while ensuring that their values fall within the range of the maximum and minimum values observed for moe and strain. How can I address this problem?
The main challenge I'm encountering is that I'm only familiar with generating correlated random variables when both distributions are normal. I tried applying the same technique used (Cholesky decomposition in particular) for normal distributions for these variables, but ended up with numerous negative values for the generated variables (a negative moe and strain does not make sense in the context on the tension tests I did). I suspect this is due to the standard deviation being of a similar magnitude as the mean. If I remove the negative values manually then the correlation coefficient is not maintained.
