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- Apr 14, 2013

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The variable $ Y $ denotes the amount of money that an adult person gives out for Christmas presents.

The distribution of $ Y $ depends on whether the person is employed ($ E = 1 $) or not ($ E = 0 $).

It holds that $ P (E = 1) = p $, i.e a randomly selected person is employed with probability $ p $.

We have the following

\begin{align*}&E(Y\mid E=1)=\mu_1 \\ &V(Y\mid E=1)=\sigma_1^2 \\ &E(Y\mid E=0)=\mu_0 \\ &V(Y\mid E=0)=\sigma_0^2 \\ &E(Y)=\mu=p\mu_1+(1-p)\mu_0 \\ &V(Y)=\sigma^2=p\sigma_1^2+(1-p)\sigma_0^2+G\end{align*}

where \begin{equation*}G=p(\mu_1-\mu)^2+(1-p)(\mu_0-\mu)^2\geq 0\end{equation*}

A research institute would like to estimate $\mu $ based on a $ n $-sized sample. The parameter $ p $ is known to the institute. Two employees of the institute, A and B, discuss the procedure.

- A suggests questioning $n$ randomly selected people and using their average spend as an estimate for $\mu $.

- B proposes to separately survey $ n p $ employed persons and $ n (1 - p) $ unemployed persons, and then use the estimator \begin{equation*} \overline{Y}_B = p \overline{Y}_1 + (1-p) \overline{Y}_0 \end{equation*} $ \overline {Y}_1 $ and $ \overline{Y}_0 $ are the average spend of the employed and non-employed persons, respectively. For the sake of simplicity, we assume that $ n p $ and $ n (1 - p) $ are integers.

If I understand correcly the proposition of B, we have a sample of soze $n$ with $np$ employed and $n(1-p)$ unemployed. $Y_{1i}$ is the answer that the employed perosn $i$ gives and $Y_{0i}$ is the answer that the unemployed person $i$ gives. We calculate the mean of what the employed people spend, according to the survey, and we define that average $\overline{Y}_1$, i.e. $ \overline{Y}_1=\frac{1}{np}\sum_{i=1}^{np}Y_{1i}$. Respectively, it holds that $ \overline{Y}_0=\frac{1}{n(1-p)}\sum_{i=1}^{n(1-p)}Y_{0i}$.

Adding these two results multiplied by the respective possibility we get the average of all people.

Have I understood that correctly?

I want to calculate the expected values and the variances of the estimates of A and B.

How could we do that? Could you give me a hint?