$n$ machines in total (starting working at the same time), working time of each is i.i.d. $Exp(\lambda)$. How to calculate the expectation of the number of broken machines at time $t$? (If broken, they stay broken forever)

ps: how to calculate the fraction of time when there are at least one broken machines by time t if repeating this process for infinite times? Can I use the expected time until the $1st$ broken down here?

  • $\begingroup$ Are you assuming they all start out working, and, once broken, they stay broken forever? $\endgroup$ – Michael Apr 11 at 2:19
  • $\begingroup$ Yes. I will revise the original question. $\endgroup$ – cxxu96 Apr 11 at 2:50
  • $\begingroup$ what is the probability of one machine be broken at time $t$? $\endgroup$ – pointguard0 Apr 11 at 2:53
  • $\begingroup$ Any machine will break down after a random time ($Exp(\lambda)$), so it might be $1-exp(-\lambda t)$? $\endgroup$ – cxxu96 Apr 11 at 2:59
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    $\begingroup$ So you want to use indicator functions by defining the number of broken machines at time $t$ as $B(t) = B_1(t) + ... + B_n(t)$ where $B_i(t) \in \{0,1\}$ is for machine $i$. $\endgroup$ – Michael Apr 11 at 3:09

The probability that any given machine has failed by time $\ t\ $ is $\ p=1-e^{-\lambda t}\ $. If the machine failures all occur independently of each other, then the number $\ F_t\ $ which have failed by time $\ t\ $ follows a binomial distribution: $$ \mathbb{P}\left(F_t=j\right) = {n \choose j} p^j\left(1-p\right)^{n-j}\ .$$ So the expected number of machines broken at time $\ t\ $ is the mean of this distribution, $\ n p\left(1-p\right) = n\left(1-e^{-\lambda t}\right)e^{-\lambda t}\ $.

As I understand the question in the ps, as now clarified, its answer is $\ \mathbb{P}\left(T_1\le t\right)\ $, where $\ T_1\ $ is the time when the first machine breaks down. This can be obtained from the identity $$ \mathbb{P}\left(T_1\le t\right)=1-\mathbb{P}\left(F_t=0\right)= 1-e^{-n\lambda t}\ . $$

  • $\begingroup$ Thanks! The question in the ps is about the fraction of time when there are more than one broken machines if repeating this process for $n \rightarrow \infty$ times. $\endgroup$ – cxxu96 Apr 11 at 14:42

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