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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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Hints:

You'll need your probability density function to have total integral 1:

$$ \int_0^{40}f(x)dx=1 $$ So, you'll have to find the right constant to stick in front of $(10+x)^{-2}$.

Then, to calculate the probability that the lifetime is less than 6, you simply integrate $f(x)$ from 0 to 6.

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Oh wow! So that was what proportional meant. Thanks! –  John Chang Dec 12 '12 at 2:08
    
Yep, any time we say "$f(x)$ is proportional to $g(x)$" we just mean $f(x)=Cg(x)$ for some constant $C$. –  icurays1 Dec 12 '12 at 2:10
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