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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?

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