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Expected expectation:TIFR 2019 GS Part A, Problem 20

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
Topic
Statistics
Difficulty Level
Moderate
Suggested Book

An Introduction to Probability and Statistics, 2ed Paperback – 2008

Start with hints

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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