Let $X_i$ and $Y_i$ two random variables with parameter $\alpha_i$ and $\beta_i$.
Let $m>0,c,b$ a positive numbers. We define $Z$ a random variable by
$$ Z= \sum_{i=0}^{m}(\max\{\log(a+cX_i),\log(a+cY_i)\}) $$
I would like to compute the probability
$$ P(Z<\theta)=P(\sum_{i=1}^{m}(\max\{\log(a+cX_i),\log(a+cY_i)\})<\Theta). $$ Can simplify to $$ P(Z<\theta)=P(\sum_{i=1}^{m}\log(\max\{(a+cX_i),(a+cY_i)\})<\Theta). $$ $$ P(Z<\theta)=P(\log(\prod_{i=1}^{m}\max\{(a+cX_i),(a+cY_i)\})<\Theta). $$ $$ P(Z<\theta)=P((\prod_{i=1}^{m}\max\{(a+cX_i),(a+cY_i)\})<e^{\Theta}). $$ Does this simplification possible.