What would be the appropriate way the calculate the probability $aX \leqslant Z \leqslant bY + cX$ where $X,Y,Z$ are all independent exponential variable with parameter ${\lambda _1},{\lambda _2},{\lambda _3}$ and $a,b,c$ are all positive number.
Since $Z$ has been squeeze between $aX$ and $bY + cX$, I guess we could use some the difference between CDF but I cannot wrap my mind around the case where $a>c$ ?
Please help me with this
Thank you for your enthusiasm !
$$P(t \leq Z \leq s)=e^{-\lambda_3 t}-e^{-\lambda_3 s}.$$ So $$P(aX \leq Z \leq bY+cX)=E(e^{-\lambda_3 aX}-e^{-\lambda_3 (bY+cX)}).$$ This is equal to $$\int_0^{\infty} \int_0^{\infty} (e^{-\lambda_3 ax}-e^{-\lambda_3 (by+cx)}) \lambda_1 e^{-\lambda_1 x} \lambda_2 e^{-\lambda_2 x} dxdy.$$