No one crime has the same circumstances, especially crimes with random victims. Any crime it's different. Psychopathic killers are different and have different fantasies and different methods. They can fantasize with young women, old women, with children, man, or just want to murder someone and killing anyone when they have the opportunity. Most killer on trails are killers on strangers. Sometimes they rape the victims with their body, sometimes rape with other things for not leave DNA, sometimes they just tell to victims to undress, humiliate them, or tell the victims to made things to one another if there is more than one victim. Sometimes they were not seen by anyone, so they don't need to tell the police they were there, etc.
I don't remember the names but there was one guy who kidnapped, raped and killed a children in 5/10 minutes who just who had only moved a few meters away from her mother. There is another case where a man pushed a teenager girl from a height in a shopping center (I don't remember very well the story but I think that it was happened). One guy went to the police and said he saw the killer and he described him. The police spent years looking for that guy and never found him. Until a new set of eyes in the investigation that thought they should investigate that man who inserted in the investigation to describe the killer. They found out he was the killer.
The killer of murder April Tinsley, she was kidnapped, raped and killed. The killer spent years taunting the police. When he was caught, he explain it was random and he just planned to kidnap a child, but he had not seen April before abducting her. He also explained he just didn't commit more crimes because the opportunity never came but he wanted to commited. Sarah Everard case in London - totally random victim. The Killer of Joenelle Matthews inserted himself in the investigation and talked with the police. The kidnapper and raper of Jayme Closs. He entered her house, killed her family and kidnapped and raped her for some time. Fortunately, she escaped with life. Police and profilers were totally wrong: they said it was to be someone who knew her family to murder all the family, etc. It was random . He just saw her in a bus stop. Etc.
I have to go to bed but then I'll see tomorrow if I remember anything else.