Break a stick of unit length at a uniformly chosen random point. Then take the shorter of the two pieces and break it again in two pieces at a uniformlychosen random point. Let X denote the length of the shortest of the final three pieces. Find the density of X."

The density of X which is the smallest piece is 1/8. This is explained below

Step-by-step explanation:

Since we are solving a problem of unit length,

There can be two events when choosing the break point, it can either lie in [0,1/2) or [1/2,1]. Assume y to be the shorter distance between an endpoint and the breakpoint.

Breaking it further, There can be another two events when choosing the break point, Half of Half. it can either lie in [0,1/4) or [1/4,1/2]. Assume X to be the shorter distance between an endpoint and the breakpoint(Second breaking)

Since we are interested in the length of the smaller piece of the three pieces, the domain of x lies in [0,1/4]

in the first case, the expected length is x*1/4

in the second case the expected length is (1/2-x)*1/4

We add the expected lengths and integrate the sum between 0 and 1/4, and that would give us 1/8.

Since these second two(Breaking the first piece into two again) events exhaust all possibilities of sample space, we have the answer as 1/8.