# Suggesting a method of moments estimator for the chance that some event happens

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 :)

• @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 – Noy Dec 4 '18 at 14:35
• Sorry. I misread as 50 weeks. You are right for using 50/(7*24). – user158565 Dec 4 '18 at 14:37