Let T be a random variable that is the time to failure (in years) of a certain t

Question

Let T be a random variable that is the time to failure (in years) of a certain type of
electrical component. T has an exponential probability density function

with beta = 2 years,

(a) Compute the probability that a given component will fail in 5 years or less.

(b) A laboratory uses 10 of these components. Let X be the number of components
out of the 10 that have failed in 5 years or less. Compute the probability that 6
components have failed in 5 years or less.

(c) For the same laboratory, compute the probability that all 10 components have failed
in 5 years or less.

Explanation / Answer

f(x) = 1/2 e^(-x/2)... x>0

a) P(x<=5 ) = integration from 0 to 5 (f(x)dx)

= 0.917

b) p = 10C6 0.917^6 0.082^4 =0.056

c) p = 10C10 0.917^10 = 0.420

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