GATE CS 2014 Set 1 — Question 12
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Engineering Mathematics → Probability & Statistics → Continuous Distributions
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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 ________ .
Correct answer
0.24 to 0.27
Solution
Let the stick be represented by the interval . We break the stick at a point chosen uniformly at random from the interval . The probability density function (PDF) of is:When the stick is broken at , it is divided into two pieces of lengths and . Let be the random variable representing the length of the shorter piece. Therefore, is defined as:We can express as a function of :Y = \begin{cases}
$ X & \text{if } 0 \le X < 0.5 \\$
$ 1 - X & \text{if } 0.5 \le X \le 1 $
\end{cases}To find the expected length of the shorter stick, , we integrate over the probability density of :Since , we split the integral at the boundary point :Evaluating the two integrals separately:
1.First integral:
2.Second integral:
Adding the two parts together:Thus, the expected length of the shorter stick is .Continue learning with Success Tracker
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