5 deleted 33 characters in body

$\max{\pi_0 W_{opt}=\arg {\max(\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha }$ \)}$subject to$\textrm{s.t.}\quad \quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi$\ We should find analytically the best optimal$W >0$when which maximize the first equation subject to the second equation, where$F( \cdot )$is probability comulative distribution function , (CDF), and$L_0$and$L_1$are positive random variables.$\xi$,$\pi_0$,$\pi_1$are constant. Also,$\pi_0, 0<\pi_0, \pi_1>0$pi_1<1$ and $\pi_0 + \pi_1 =1$. All variables are real. Further, if needed, we can assume that, for example, $F_{L_0}$ and $F_{L_1}$ such as below ($\lambda$ L_0$and$\mu$are constant:$F_{L_0}(\alpha) &=& 1-e^{-\mu \alpha} \quad \alpha \geq 0.F_{L_1}(\alpha) &=& 1-e^{-\lambda \alpha} \quad \alpha \geq 0L_1$may have Erlang or exponential distribution.$

4 added 89 characters in body

$\max{\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha }$ \

$\textrm{s.t.}\quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi$ \

We should find the best $W$ W >0$when$\xi$is constant,$F( \cdot )$is probability distribution function, and$L_0$and$L_1$are positive random variables.$\xi$,$\pi_0$,$\pi_1$are constant. Also,$\pi_0, \pi_1>0$and$\pi_0 + \pi_1 =1$. All variables are real. Further, if needed, we can assume, for example,$F_{L_0}$and$F_{L_1}$such as below ($\lambda$and$\mu$are constant:$F_{L_0}(\alpha) &=& 1-e^{-\mu \alpha} \quad \alpha \geq 0.F_{L_1}(\alpha) &=& 1-e^{-\lambda \alpha} \quad \alpha \geq 0. $3 deleted 63 characters in body; deleted 2 characters in body $$\textrm{Problem}\qquad\qquad \begin{cases}\max{\pi_0 \max{\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha }, \\ \textrm{s.t.}\quad \textrm{s.t.}\quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi'' <\xi \end{cases}$$ We should find the best$W$when$\xi$is the variableconstant, and$F( \cdot )$is probability distribution function, and$L_0$and$L_1$are random variables. Also,$\pi_0, \pi_1>0$and$\pi_0 + \pi_1 =1$. Further, if needed, we can assume, for example,$F_{L_0}$and$F_{L_1}$can be such as below: \begin{eqnarray} F_{L_0}(\alpha)$F_{L_0}(\alpha) &=& 1-e^{-\mu \alpha} \quad \alpha \geq 0. \label{PDF0}\ F_{L_1}(\alpha) 0.F_{L_1}(\alpha) &=& 1-e^{-\lambda \alpha} \quad \alpha \geq 0. \label{PDF1} \end{eqnarray}\$

2 Tex
 
1