The question is based on an engineering problem and, due to my lack of knowledge in statistics, I found it difficult to express the question clearly in maths language.
The problem is formulated like this: 1) Imagine there is an origin, and a 1D coordinate system (say, L);
2) Along this 1D axis (L), there is a 'rigid' rod attaching to the origin, and its length is a random variable (always positive) and follows a PDF p1.
Question 1: for each given point on the axis L, what is the chance of that point belonging to the thread.
My solution to Q1: for a given coordinate x, for it to 'fall' into the thread, the thread length (a) has to be equal or greater than x. Therefore, the chance of a point x belonging to that rod is equal to the integration of that PDF, from x to infinity.
Can anyone please confirm if my solution above is correct?
If so, the question 2 is as follows: if the near end of the rod no longer attaches to the origin, but the location of it is also a random variable, which follows, say, another, PDF p2. Then, for each given point on the axis L, what is the chance of that point belonging to the rod?
I am so confused by this question 2, as I have no idea what I should do with the potential interplay between these two random events (i.e. the length of rod and the location of near end of the rod). I wonder if anyone here can help me with this, or at least let me know what kind of 'theory' I should look into for solving this problem?