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I have three records. the records means intervals,

  1. $A: [1, 5]$

  2. $B: [2, 6]$

  3. $C: [4, 6]$

A, B and C are three humidity sensors. Value of A is between 1 to 5, B is between 2 to 6, and C is between 4 to 6

Assume the humidity data is uniform distributed(uniform distribution). I'd like to know which sensor would give the minimal value(lowest humidity), with probability.

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    @Henry I've edited the question2012-11-02

1 Answers 1

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Clearly the minimum falls in the ranges $[1,2)$ or $[2,4)$ or $[4,5]$ so the probabilities A, B or C give the minimum respectively are

$\int_{a=1}^2 \frac{da}{5-1}\cdot1\cdot1 + \int_{a=2}^4 \frac{da}{5-1} \cdot \frac{6-a}{6-2} \cdot 1 + \int_{a=4}^5 \frac{da}{5-1}\cdot \frac{6-a}{6-2}\cdot\frac{6-a}{6-4}$

$0 + \int_{b=2}^4 \frac{5-b}{5-1}\cdot \frac{db}{6-2}\cdot 1 + \int_{b=4}^5\cdot \frac{5-b}{5-1} \frac{db}{6-2}\cdot\frac{6-b}{6-4}$

$0 + 0 + \int_{c=4}^5 \frac{5-c}{5-1}\cdot \frac{6-c}{6-2}\cdot\frac{dc}{6-4}$

and if you calculate these they should add up to $1$.

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    I think I get it now, and you are right.2012-11-06