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I am working on a problem that requires me to find the UMVUE of $Pr[X=0]=e^{-\lambda}$.

There are several issues that I am having so I would really appreciate your help.

1), What does it mean to be an "estimator" of a probability? I understand that estimators can estimate parameters such as the mean or variance of a distribution, but I do not quite get the intuitive meaning of this problem. . .

2), The notes that I am looking at mentions

$$\Bbb{I}_{\{X_1=0\}}$$

is a "natural choice" of an unbiased estimator of $Pr[X=0]$.

I am thinking that because I don't understand intuitively what is going on I am not seeing why an indicator is an estimator.

The notes further proceeds to simplifying

$$E[\Bbb{I}_{\{X_1=0\}}|\sum_{i=1}^n X_i=x]$$

which leads to

$$\left( \frac{n-1}{n} \right)^x$$

I understand the rest of the algebraic portion, but because I don't know why it started off like this it does not stay in my head.

Thank you for your help.

hyg17
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  • Because $E\left[\mathbb I_{{X_1=0}}\right]=P(X_1=0)=e^{-\lambda}$. The probability is a function of $\lambda$ which is unknown, so you are estimating it. – StubbornAtom Jun 29 '19 at 21:10
  • So, let me make sure that I understand this. $\Bbb{I}{X_1=0}=1$ when $X_1=0$ and $=0$ when $X_1\ne 0$ so under that condition $E[\Bbb{I}{X_1=0}]=\sum_{X}\Bbb{I}_{X_1=0}Pr[X=x]$. So, if $X_1=0$ then $\because = 1Pr[X_1=0]$ and when $X_1 \ne 0$ then $\because = 0*Pr[X_1 \ne 0]$ – hyg17 Jun 29 '19 at 22:09
  • I see that numerically it is true. However, why would it be "natural" to think that this is the estimator? I mean, this means that the probability is either 1 or 0 right? I do not find that to be natural at all. . . – hyg17 Jun 29 '19 at 22:12
  • Not numerically, it is exactly true. To repeat, perhaps the most trivial unbiased estimator of a parametric function like the probability $P(X\in A)$ is the indicator of the event $X\in A$, since $E(I_{X\in A})=1.P(X\in A)+0.P(X\notin A)$. – StubbornAtom Jun 29 '19 at 22:56
  • Hmm... I will just accept the algebraic part and hope that I spend enough time in stats that someday I understand. – hyg17 Jun 30 '19 at 16:47
  • https://math.stackexchange.com/q/2689415/321264 – StubbornAtom May 19 '20 at 14:38

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