We suppose that X$\sim $(μ,Σ) with mean-vector μ=$(1.5,1.5)^t$, covariace matrix Σ=$\binom {1.5\quad 1.2}{1.2\quad 1.5}$ and $\boldsymbol{Y}=e^\boldsymbol{X}$. I would like to sample from a bivariate lognorm using R.
By using the command lnorm.rplus i can sample from multivariate lognorm.But my problem is that i can't determine the arguments for meanlog and varlog.
Can anybody help me find the arguments ?
Hints:
$$E(e^{X_i}) = M_{X_i}(1)$$
$$E((e^{X_i})^2) = E(e^{2X_i}) = M_{X_i}(2)$$
$$Cov(e^{X_1}, e^{X_2}) = E(e^{X_1+X_2}) - E(e^{X_1})E(e^{X_2})$$
where
$$E(e^{X_1+X_2}) = M_{X_1,X_2}(1,1)$$
$$M_{X_i}(t) = E(e^{tX_i}) = e^{\mu_it+\sigma_i^2t^2/2}$$
$$M_{X_1,X_2}(t_1, t_2) = E(e^{t_1X_2+t_2X_2}) = e^{\mu_1t_1+\mu_2t_2 + \frac{1}{2}(\sigma_1^2t_1^2+\sigma_2^2t_2^2+2\rho\sigma_1\sigma_2t_1t_2)}$$
As mentioned on this page, you must perform $\log$ transformation on the elements of mean vector derived from above equations before passing them as an argument to the function you mentioned. Not sure if the elements of the covariance matrix also need to be log-transformed before passing them as argument to that function.
