Re: Probability with line E(X).


"mina_world" <mina_world@xxxxxxxxxxx> wrote in message news:gbflc4$og9$1@xxxxxxxxxxxxxxxxxxxxxxxx
Hello teacher~

There is a line with length 1.
Namely, [0, 1].

Choose a point c in [0, 1].
so, there is two separate segments.
(of course, if c = 0 or c = 1, Consider that
fictitious two separate segments with length 0.)

Let X be the longest length among two separate segment.

Find E(X).

--------------------------------------------------------
Let Y be the length of [0, c].

so, x = 1 - y , (0 <= y < 1/2)
= y , (1/2 <= y <= 1)

f(y) = 1 on [0, 1]. (p.d.f of Y.)

E(X) = int{1/2 to 1} x.f(x) dx

Hm... I need f(x). but I don't know well.
so, I need your advice.


You break it at point c.

There are two cases, the first part longer (c>0.5) and the second part longer (c<0.5)

If c>0.5, then all you know is that it its length is uniformly distributed over [0.5,1.0] so its expectation is 0.75. By symmetry, if the second part is longer (c<0.5), exactly the same argument applies.

So the expectation is 0.75 either way, and so this is the total expectation.






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