Htykuut

Following Expected Value of Maximum of Two Lognormal Random Variables . I made an additional assumption, that $X,Y$ are jointly normal with parameters given above.

I tried to derive the simmilar result by first finding the density function of $max(X,Y)$, and I have the following CDF: $P(max(X,Y) < m ) = P(X < m, Y < m) = Phi(frac{(log(m) - mu)} {sigma}, frac{(log(m) - nu)} {tau})$, where $Phi$ is the CDF of multivariate standard normal.

Then, I computed the density function which is $phi_{max} = frac{1}{sigma times m}phi_1(frac{(log(m) - mu)} {sigma}) + frac{1}{tau times m} phi_2(frac{(log(m) - nu)} {tau})$, where $phi_1$ and $phi_2$ are marginal distribution functions of joint multivariate normal, which in turn are just densities of related normal.

Since I know, the distribution function at each $m$, then I can compute its expectation. I form $E(max(X,Y)) = int_0^{infty}(m times phi_{max} dm) = int_0^{infty} m times frac{1}{sigma times m}phi_1(frac{(log(m) - mu)} {sigma}) dm + int_0^{infty} m times frac{1}{tau times m} phi_2(frac{(log(m) - nu)} {tau}) dm = int_0^{infty} frac{1}{sigma}phi_1(frac{(log(m) - mu)} {sigma}) dm + int_0^{infty} frac{1}{tau} phi_2(frac{(log(m) - nu)} {tau}) dm = exp(mu + frac{sigma^2}{2}) + exp(nu + frac{tau^2}{2})$. Seemingly, I have made mistake when considering marginal distributions but cannot see where exactly.

Where I have got a flaw in my derivations?