expected value involving probability of inequality random variables.

Question

I have a question, not sure if this can be solved by calculation or Monte Carlo method For random variable G2+G2*min(2/G2-1,G1) where G1, G2 are indenpendt, G1~Lognormal(mu1,cigma1) G2~Lognormal(mu2,cigma2)

Find the expected value and variance.

I have simplified to this r.v. is 2 with prob(2/G2-1G1)

Thanks for time.

Answer 1

Using $G_i=\mathrm e^{X_i}$ where the distribution of $X_i$ is normal, this is $G=\min(2,H)$ with $H=\mathrm e^{X_2}(1+\mathrm e^{X_1})$. Assuming the density of $H$ is $f_H$, $G$ has density $f_H$ on $(2,\infty)$ and a mass $P[H\leqslant2]=\int\limits_{-\infty}^2f_H$ at $2$. Now one is supposed to compute $f_H$. What is stopping you?