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Sagot :
The expected length of the middle part of the stick is 1/3
We know that the Symmetry Principal,
When n points are drooped on a unit segment, the distributions of lengths of the so obtained n + 1 pieces are all the same.
Let the two points have coordinates x and y. Then the expected value of the middle segment is:
[tex]\int\limits^1_0 \int\limits^1_0 {x-y} \, dx~ dy = \frac{1}{3}[/tex]
Let L be the expected value of the length of the middle part of a stick. Since, according to the Symmetry Principle, the lengths of all three segments have the same distribution, their lengths have the same expected value, viz., L.
But in any experiment, the lengths of the three pieces add up to 1.
⇒ 3L= 1
⇒ L = 1/3
Therefore, the expected length of the middle part of the stick is 1/3
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