1. order statistics

say i have three machines connected in parallel. the life expectancy of each machine is exponentially distributed with beta value of 10. The system is in the working state if any one of the connected component is working.

Find the probability of the system operating for more than 5 hours.

2. Ill assume the lifetime of each machine is independant of the others.

$X_{1,2,3} = \text{Life of machine i}$

$P(\text{system dead at 5 hours}) = P(X_1 < 5,X_2 <5,X_2 < 5,X_3 < 5) = P(X_1 < 5)P(X_2 < 5)P(X_3 < 5)$

$=(1-e^{-5\lambda}))^3$

I think Lambda is 1/beta. in any case, just use the CDF as you ahve been taught it.

$P(\text{system lives more than 5 hours}) = 1-P(\text{system dead at 5 hours})= 1-(1-e^{-5\lambda}))^3$

3. say if each machine's are having different beta values

machine1 --> beta1 = 5
machine2 --> beta2 = 6
machine3 --> beta3 = 7

Now is i this a order statistic sum?

4. you should be able to generalise what i wrote, have a go

5. i heard that to apply order statistics the distributions should be the same now if the beta values are different then the distributions will not be the same. Am i correct

So in this case you cannot apply order statistics principles you have to go in the general way

am i correct?

6. if by "order statistics principles" you mean using the binomial distribution to get a quick answer, then you are correct.

7. no i did not mean for binomial distribution.

X1 - distribution of macine 1
X2 - distribution of machine 2
X3- distribution of machine 3

each machine's life is exponentially distributed.

But since each machine's beta values are not the same order stat principles cannot be applied in this case. Am i correct ?