I am attempting a question for personal study from my text book but cant seem to get the right method. I am unsure how to even start it. The question is:
You wish to fence off a rectangular paddock on one side of a river running through your property in a straight line. No fence is required along the side of the paddock formed by the river. The fence you will use is rolled up in a shed, and you are at the moment not quite sure how long it is. However, you are certain that it is between 3 and 5 km long, and your uncertainty regarding its length can be represented by a parabolic probability density function which tapers off to zero at 3 and 5 kms.
a) Find and sketch the probability density function of Y, the area of the largest paddock you will be able to fence off.
b) Find the expected value, mode and median of Y. Then illustrate these three quantities in the figure in (a).
I understand that they may be a constant involved and we can integrate the function and let it be equal to 1 to solve for the constant. I think b would be much easier if I had an answer for a.
Any help would be appreciated
First, you need to write down a probability density for the length of fence you have. Clearly, it must be a quadratic polynomial with negative leading coefficient, with roots at $3$ and $5$.
Second, for some fixed known length of fencing, you need to find out what rectangular shape maximizes the area you can enclose with that length of fence. Then express the area in terms of this length.
Third, compute the density function for the area via transformation of the random variable representing the length of fencing.