I have not made this study yet, but I am fairly sure of what the results will be before hand since I have lead the life I have.
What is the most prevalent cause of persons recognizably of the inappropriate gender being in a gender designated restroom?
First Cause Service personnel cleaning or servicing the restroom where the customer ignored warning signs.
Second cause An intoxicated person who was oblivious to their surroundings.
What is the most common reason for young children being surprised or upset by a person of the inappropriate gender being in a gender designated rest room.
First Cause Lack of parental supervision and accompaniment/Mistake by child for that reason.
Second Cause Intoxicated parent that directed them to the wrong bathroom
Third Cause An intoxicated person who was oblivious to their surroundings.
While it is true that transsexuals and other trangender folks can be the intoxicated offender, it was the intoxication that was the primary cause, not the garb or anatomy.
I live in an area where there are and have been children assaulted by people in restrooms, and in each case, the parent was not with the child when it happened. To my knowledge, NONE of the assaulters was in the clothing of an inappropriate gender for the restroom they chose as their "seat of power".