Question

A stick of length l is broken at a uniformly chosen random location. We denote the length of the smaller piece by X. (a) Find the cumulative distribution function of X. (b) Find the probability density function of X.

Answer 1

a)
`\phi (x) = a/(I/2)`

b)
`f(x) = 2/I`

Explanation:

a) Lets denote
`\phi` the cumulative distribution function of X. Note that for any value a between 0 and I/2, we have that
`\phi(a)` is the probability for the stick to be broken before the length a is reached following the stick from one starting point plus the probability for the stick to be broken after the length I-a from the same starting point. This means that
`\phi(a) = (a+a)/I = 2a/I = a/ (I/2)`

b) Note that, as a consecuence of what we calculate in the previous item, X has a uniform distribution with parameter I/2, therefore, the probability density function f is

f(x) = 1/(I/2) = 2/I