I would like to solve the following excercise. Any help is appreciated.
90% of the fish in our pond are males, the rest are females.
The length of the males are:
$X+5$ inches, where $X\sim exp(1)$
The length of the females are:
$Y+8$ inches, where $Y \sim exp(2)$
What is the probability that a fish whose length is $x$ is male, and how can we guess the sex of the fish from their length if we want that our guess is right with the biggest possible probability?