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Let $X_i$~ $\text{Pois}(\lambda)$ be the number of breakdowns a certain ATM machine experiences in the $i^{th}$ week. $\implies$ Let $ \{X_i\}_{i=1}^n$ be iid of the number of breakdowns the machine experiences in $n$ different weeks.

You managed to withdraw cash from the machine. Suggest a method of moments estimator for the chance that the machine will work properly within the next 50 hours (no breakdowns for 50 hours), based on the weekly number of breakdowns $\hat{\lambda}_{MLE}$.

Hey everyone. I am pretty new to statistics, and therefore to this community, so excuse me if I break any rule or don't understand simple concepts...

I had already computed $\hat{\lambda}_{MLE}$ and found that $\hat{\lambda}_{MLE}=\bar{X}$. Thus, an efficient method of moments estimator (that is unbiased) of $\lambda$ would be $\hat{\mu}_1=E[X_i]=\lambda=\bar{X}$.

I don't understand though how to find an efficient method of moments estimator for a chance that some event happens, like no breakdowns for 50 hours.

I thought of defining a random variable $Y$~$\text{Poi}(\frac{50}{7\cdot 24}\lambda)$ and computing $P(Y=0)$ but this is not what we are required to do... I would be happy to get your help on how to find a proper method of moments estimator for this chance.Thank you in advance :)

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  • $\begingroup$ @user158565 $\lambda$ is the expected number of breakdowns in a week so $\frac{1}{168}\lambda$ would be in an hour. But should I just compute $P(Y=0)$? The question asks us to find an estimator based on moments.. I'm not sure how to do it $\endgroup$ – Noy Dec 4 '18 at 14:35
  • $\begingroup$ Sorry. I misread as 50 weeks. You are right for using 50/(7*24). $\endgroup$ – user158565 Dec 4 '18 at 14:37

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