Suppose that a system contains a certain type of component...

Suppose that a system contains a certain type of component whose time, in years, to failure is given by T . The random variable T is modeled nicely by the exponential distribution with mean time to failure ? = 5. a) what is the probability that a component still is functioning at the end of 8 years (its life is more than 8 years)? b) what is the probability that a component lasts at most 8 years? c) If 5 of these components are installed in different systems, what is the probability that at least 2 are still functioning at the end of 8 years? d) If 5 of these components are installed in different systems, what is the probability exactly 3 last at most 8 years?

Transcript

00:01 Hi, i'm david and i'm here to help you answer in your question.

00:03 Now let me bring up your question here.

00:07 In the question we discuss about the exponential distribution and in the question c and d we will need to use the binomial.

00:15 So let me remind you that if we have the x followed by exponential with the mean beta and then the fx it will equal to the 1 over beta and then the fx is equal to the 1 over beta and then then e to the power minus x over beta for the x greater than 0.

00:39 Also the cumulative distribution of the x equal to the probability x -small equal to the x.

00:46 This is just equal to the integral 1 over beta, e to the minus x over beta, from 0 to the x, the x.

00:56 We get this one.

00:57 If we do it, we should get this one, equal to 1 minus e to the power minus x over beta for the x greater than 0.

01:06 Now in this question we're given that the x it will follow by exponential with the beta equal to 5.

01:15 And the question a asking us to find the probability that it will still function at the end of the 8 years means that x will be greater than 8.

01:27 So by common moment we can do 1 minus probability the x.

01:30 Is more equal to 8.

01:33 And the probability that equal to the cdf, f of the 8.

01:38 Therefore we have the 1 minus each the power.

01:42 I use the formula here.

01:44 So we have the 1 minus 8 over 8 over 5 here because that would be the better.

01:55 And if we do the calculation, we will have minus 8, 5 e power on the 8.

02:02 Answer equal to 0 .2 02.

02:07 That will be the answer for the 8.

02:09 So now for the b, want to find the proper pitted that will last at most eight years.

02:16 So means that x will be is more equal to 8.

02:19 This one equal to the 1, it will exactly equal to the f of the 8.

02:25 So we get equal to 1 minus e to the power minus 8 out of 5.

02:30 So we just equal to the 1 minus the answer so equal to the 0 .7 9 8 and in the question c we are given that n equal to the 5 component and want to find the proper property that unless 2 are still functioning at the end of the 8 years so we'll call be the proper premium survival and proper printing the x greater than 8 we found that the answer will equal to the 0102 in the part a...

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