GATE CSE 2014 Set 1 | Question: 2

Question

Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .

Comments

For proper solution please refer --> https://math.stackexchange.com/questions/350679/we-break-a-unit-length-rod-into-two-pieces-at-a-uniformly-chosen-point-find-the/350863#350863

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why we are taking length of shorter stick as 0, won’t it be 0.01? as we must have break from a point so how it can remain unbroken(as we are considering shorter stick size is 0) my approach was like 0.01 for shorter stick min length and 0.49 for shorter stick max length( 0.50 will create both equal hence not any will be smaller and from 0.51 we will start getting this bigger) so b+a/2=0.01+0.49/2=0.25

can someone please correct me if I am missing something?

Answer 1

The length of the shorter stick can be from $0$ to $0.5$ (because if it is greater than $0.5,$ it is no longer a shorter stick).

 This random variable $L$ (length of shorter stick) follows a uniform distribution, and hence probability density function of $L$ is $\dfrac{1}{0.5-0}= 2$ for all lengths in range $0$ to $0.5$

Now expected value of $L = \int_{0}^{0.5} L*p(L) dL = \int_{0}^{0.5} L*2 dL = 2*\left[\dfrac{L^2}{2}\right]^{0.5}_0 = 0.25$

Comments

Length of the shorter stick can be from 0.5 to 1 OR  0 to 0.5. so even if you take range 0.5 to 1 you will get same answer.

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But in expectation we consider both the scenarios, so why here only one part is being considered.

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Both scenarios are identical so expectation will not change.

if you consider scenario 1 then you get 0.25, if you consider scenario 2 then you get 0.25 .

expecation = (scenario1 + scenario 2 )/2 = (0.25 +0.25) /2 = 0.25

Answer 2

Answer = $0.25$

Let $X$ = Length of Shorter Stick

$\implies 0<X<\frac{1}{2}\implies X$ follows Uniform Distribution over $(0,\frac{1}{2})$

We know, In uniform distribution, the mean (first moment) of the distribution is:

${\displaystyle E(X)={\frac {1}{2}}(b+a).}$   Link : Continuous uniform distribution - Wikipedia

Hence, the expected length of the shorter stick is = $Mean$ = $E[X] = \frac{b+a}{2} = \frac{0+\frac{1}{2}}{2} = \frac{1}{4} = 0.25$

Comments

Beautiful
Short & simple approach

Answer 3

Expectation in case of a Uniform Random Variable = $\frac{b-a}{2}$

https://www.ucd.ie/msc/t4media/Uniform%20Distribution.pdf

As shorter length stick is specified in question, take a=0 and b=0.5

$\frac{0.5}{2}$ = ¼ = 0.25

Comments

@ashley they have given it wrong, right one is a+b/2 not b-a/2 you can try to calculate what they have they did a mistake in calcuation so be careful