Can anyone have any suggestions how to solve this equation for $w_i$, that is, what is the solution of $w_i$? $$ \int_0^\infty e^{\Phi^{-1}(w_i)ε_i}P(r_i│ε_i )f(ε_i )dε_i=δ $$
Where, $f(ε_i)$ is the probability density function of $ε_i$, $ε_i$ can be exponential random variable, $\Phi^{-1}(w_i)$ is the inverse of the normal CDF, $P(r_i│ε_i )$ is an unknown function, and all the remaining parameters are constant. I know it may not be possible to have an exact solution. However, there might have some approximation or clever trick, or possibility to get upper and lower limit of $w_i$. Thank you I appreciate your help :)
Or, can we write the above equation as $$ \int_0^\infty e^{\Phi^{-1}(w_i)ε_i}P(r_i│ε_i )f(ε_i )dε_i=δ\int_0^\infty f(ε_i )dε_i $$ and hence by cancelling out the integrations from the both sides $$ e^{\Phi^{-1}(w_i)ε_i}P(r_i│ε_i )=δ $$ because $\int_0^\infty f(ε_i )dε_i=1$