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I need help with homework exercise, have no idea how to approach it.

Assume we have i.i.d. observations $x_1,\ldots,x_n$ of a continuous random variable $X$, taking values in $\mathbb R^+$. Define the discrete random variable $Y$ as:

$$Y=\sum_{i=0}^\infty i\{i-1\leq X <i\}$$

(a) Derive the plug-in estimator of the probability mass function for $Y$.

(b) Derive the plug-in estimator of the expectation $E(Y)$

A word about the notation, I belive that $\{i-1\leq X <i\}$ is a boolean expression taking the value $1$ when true and $0$ when false, but I'm not certain.

I am drawing a complete blank here so help or guidance would be much appreaciated.

$\bf Edit1:$ Some new thoughts thanks to the helpful comments.

$$p_Y(k)=P(k-1\leq X < k)=F_X(k)- F_X(k-1)$$

We can estimate $F_X$ with the empirical distribution:

$$\bar F_X(t)=1/n\sum_{i=0}^n1\{x_i<t\}$$

For (a) this gives us an estimate:

$$\bar p_Y(k)=\bar F_X(k) - \bar F_X(k-1)=1/n\sum_{i=0}^n1\{x_i<k\} - 1/n\sum_{i=0}^n1\{x_i<k-1\}=$$

$$=1/n\sum_{i=0}^n1\{k-1\leq x_i<k\}$$

We could use this estimator for (b) but frankly I'm not sure this is what they are after.

We can approximate; $$E(Y)=\sum_{k=0}^\infty k p_Y(k)$$ using: $$\bar E(Y) = \sum_{k=0}^\infty k \bar p_Y(k)=1/n\sum_{k=0}^\infty \sum_{i=0}^n k\{k-1\leq x_i<k\}=1/n\sum_{i=0}^n \sum_{k=0}^\infty k\{k-1\leq x_i<k\}$$

This expression feels very ,uch like a tautology to me and I'm not sure I interpreted everything correctly, for instance I'm not sure I interpreted the definition of $Y$ correctly, or that the estimators I gave are so called plug-in estimators.

So comments, confirmations or corrections are still extremely welcome.

Regards, Tobias

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How was the term "plug-in estimator" defined? – Michael Hardy Jan 18 '13 at 1:49
I see: You apply a functional to the empirical distribution and use that as an estimator of the same function applied to the population distribution. – Michael Hardy Jan 18 '13 at 3:37
@MichaelHardy. Yes, thats the definition, thou I'm not quite sure how to use it. – user25470 Jan 18 '13 at 8:46

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