What is the probability that the time until the first machine break-down is at least 5 days?

Question

A factory contains 8 machines. Any one of the machines can breakdown independently of the other machines following an exponential distribution with a mean value of 10 days. What is the probability that the time until the first machine break-down is at least 5 days?

obviously $\mu = 10$ and hence, $\lambda = 1/10$

I think we have that $P(X>5) = 1-P(X<5) = 1-(1-e^-5\times0.1)) = 0.60653$ but this is just for one machine. I'm not sure how to deal with the fact that there are 8 machines.

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Maximilian Janisch · Accepted Answer · 2019-10-05T18:30:00.807

2

Hint. Let $X_1, \dots, X_n$ be independent exponentially distributed random variables with rate parameters $\lambda_1,\dots, \lambda_n$. Then

$$\min \left \{X_1,\dotsc,X_n \right \}$$

is also exponentially distributed, with parameter

$$\lambda = \lambda_1+\dotsb+\lambda_n.$$ (Source: Wikipedia.)

So you need to calculate $\mathbb P(X\geq 5)$ where $X$ has an exponential distribution with parameter $\frac8{10}$.

edited Oct 05 '19 at 18:30

answered Oct 05 '19 at 17:59

Maximilian Janisch

13,536

So would i be correct in saying i should have $1-(1-e^(-5\times1.2))$ – ho_howdy Oct 05 '19 at 18:18
@ho_howdy That is correct. Make sure to accept my answer in case you are satisfied with it – Maximilian Janisch Oct 05 '19 at 18:19
that returns 0.00247, that's surely not the correct answer? – ho_howdy Oct 05 '19 at 18:25
@ho_howdy Sorry, I made a small error: The parameter is $\frac8{10}$ and not $\frac{10}8$... So the correct answer is $e^{-4}$. – Maximilian Janisch Oct 05 '19 at 18:31
understood. thank you for your help! – ho_howdy Oct 05 '19 at 18:33
@ho_howdy You're welcome! – Maximilian Janisch Oct 05 '19 at 18:35
Hi, i have another question regarding this. What is the distribution of the time until the first machine breaks down?
I think that the distribution of time between two break downs is also exponentially distributed. But since there are 8 machines, I am thinking it is Gamma distributed? But i am not sure.
– ho_howdy Oct 07 '19 at 11:44
@how_howdy The distribution of the time until the first breakdown is exponential and so is the distribution of the time of any given machine. For the distribution of time between two breakdowns see here: https://math.stackexchange.com/questions/115022/pdf-of-the-difference-of-two-exponentially-distributed-random-variables – Maximilian Janisch Oct 18 '19 at 17:34

What is the probability that the time until the first machine break-down is at least 5 days?

1 Answers1