This post is related to previous SE posts, could anyone please explain the final geometry steps i.e. the link between the geometry and the pdf.
- Average length of the longest segment : this part : " By rewriting this probability as area in the unit square, I get " @TCL.
@TCL gave a very clear explanation for almost the same problem with : If a 1 meter rope is cut at two uniformly randomly chosen points, what is the average length of the smallest piece? but could you please explain this part : " which is (1−3a)2(1−3a)2. Note that for these sets to be nonempty, one must have 0≤a≤1/3 0≤a≤1/3 "
- Stick of unit length is broken into three random pieces, what is the expected length of the longest piece? : this part : " A bit of geometry will give you the result " @Canardini
Another very quick reply to a very similar problem : @ Hans Parshall
- If a 1 meter rope is cut at two uniformly randomly chosen points, what is the average length of the smallest piece?" This part : It's not hard to show that they all have probability ... / our joint PDF is given by f(x,y)=6f(x,y)=6"