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Suppose a system contains certain type of component whose lifetime is T .Random variable T is modelled by exponential distribution with mean time to failure is b=5 .Probability that given component installed on system is still working after 8 years is

1)0.16  2)0.17 3)0.18  4)0.2

In above problem if 5 of these components are installed in different systems.What is probability that atleast 2 components are still functioning at end of 8 years?

1)0.4128 2)0.3645.3)0.3149 4)0.2627
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1 Answer

4 votes
4 votes
Best answer
Cumulative distribution function of exponential distribution is given by $F(x) = 1 - e^{-\lambda x}$, where $\lambda$ is the rate of occurrence of event. (https://en.wikipedia.org/wiki/Exponential_distribution#Cumulative_distribution_function )

So for example, $F(8)$ means component survives at most 8 years.

Here mean time of failure is 5 years i.e. rate parameter $\lambda$ is 1/5 = 0.2.

(a) Probability of component working even after 8 years is $1 - F(8) = 1 - (1-e^{-0.2*8}) = e^{-1.6} \approx 0.20 $

Hence option (4) is correct.

(b) Probability that at least 2 systems are working = 1 - (prob of no system working + prob of 1 system working)

Let probability of a system working after 8 years is $p$, which is 0.20 as calculated earlier.

So required probability is

$$P = 1 - \binom{5}{0}(p)^0(1-p)^5 - \binom{5}{1}(p)^1(1-p)^4 \approx 0.2627$$

Hence option (4) is correct.

2 Comments

nicely answered.thnks :)
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Will the second answer would be same for the system working even after 100 years?
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