Calculate Probability Machine Part Lifetime < 6: Integration Problem

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SUMMARY

The discussion focuses on calculating the probability that the lifetime of a machine part, modeled by a continuous probability density function (PDF) f(x) proportional to (10 + x)-2, is less than 6. The PDF is defined as f(x) = k(10 + x)-2, where k is a normalization constant. Participants clarify that k is necessary to ensure the total area under the PDF equals 1, which is a fundamental requirement for any probability distribution.

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Homework Statement

the lifetime of a machine part has a continuous distribution on the interval(0,40) with probability density function f, where f(x) is proportional to (10 + x)^(-2) Calculate the probability that the lifetime of the machine part is less than 6.



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The Attempt at a Solution

My book says that since f(x) is proportional to (10 + x), f(x)=k(10+x)^(-2). My question is why we need to put the k in the equation? Is it merely because the question says that they're proportional and so we need to set up that integration to find by how much they're proportional?
 
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