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Alexxx [7]
2 years ago
14

A stick of length l is broken at a uniformly chosen random location. We denote the length of the smaller piece by X. (a) Find th

e cumulative distribution function of X. (b) Find the probability density function of X.
Mathematics
1 answer:
kykrilka [37]2 years ago
6 0

Answer:

a) \phi (x) = a/(I/2)

b) f(x) = 2/I

Step-by-step explanation:

a) Lets denote \phi the cumulative distribution function of X. Note that for any value a between 0 and I/2, we have that \phi(a) is the probability for the stick to be broken before the length a is reached following the stick from one starting point plus the probability for the stick to be broken after the length I-a from the same starting point. This means that \phi(a) = (a+a)/I = 2a/I = a/ (I/2)

b) Note that, as a consecuence of what we calculate in the previous item, X has a uniform distribution with parameter I/2, therefore, the probability density function f is

f(x) = 1/(I/2) = 2/I  

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