Understand the problem

A stick of length $$1$$ is broken into two pieces by cutting at a randomly chosen point. What is the expected length of the smaller piece?
1. $$1 /8$$
2. $$1 /4$$
3. $$1 /e$$
4. $$1 /\pi$$

Source of the problem
TIFR 2019 GS Part A, Problem 20
Statistics
Moderate

An Introduction to Probability and Statistics, 2edPaperback– 2008

Do you really need a hint? Try it first!

Can you use uniform probability distribution in this question?
Let’s let $$x$$ denote the length of the smaller piece.
Then $$x$$ is uniform on $$[0, 1/2]$$, as the split-point $$s$$ is uniform on $$[0, 1]$$, but for split points past $$1/2$$, the “smaller piece” $$x$$ becomes $$1-s$$ instead of $$s$$. Try to calculate the expectation now.
In this case, it’s continuous uniform prabability distribution, and we use
$$E(x) = \int x \cdot p(x) ~dx$$
where $$p(x)$$ is the probability density
The probability density $$p(x)$$, defined on $$[0, 0.5]$$, is given by
$$p(x) = 2$$
(its integral over the interval is exactly $$1$$). So
you need to compute
$$\int_0^{0.5} x \cdot 2 dx.$$
So the answer you will get after completing the integral is $$\frac 14$$

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