Exercise

# Exercise 7. Expected value of d

Our estimate for the difference in proportions of Democrats and Republicans is \(d = \bar{X} - (1-\bar{X})\).

Which derivation correctly uses the rules we learned about sums of random variables and scaled random variables to derive the expected value of \(d\)?

Instructions

### Possible answers

$$\begin{eqnarray} \mbox{E}[\bar{X} - (1-\bar{X})] &=& \mbox{E}[2\bar{X} - 1] \ &=& 2\mbox{E}[\bar{X}] - 1 \ &=& N(2p - 1)\ &=& Np - N(1-p) \end{eqnarray} $$

$$ \begin{eqnarray} \mbox{E}[\bar{X} - (1-\bar{X})] &=& \mbox{E}[\bar{X} - 1] \ &=& \mbox{E}[\bar{X}] - 1 \ &=& p - 1\ \end{eqnarray} $$

$$ \begin{eqnarray} \mbox{E}[\bar{X} - (1-\bar{X})] &=& \mbox{E}[2\bar{X} - 1] \ &=& 2\mbox{E}[\bar{X}] - 1 \ &=& 2\sqrt{p(1-p)} -1 \ &=& p - (1-p) \end{eqnarray} $$

$$ \begin{eqnarray} \mbox{E}[\bar{X} - (1-\bar{X})] &=& \mbox{E}[2\bar{X} - 1] \ &=& 2\mbox{E}[\bar{X}] - 1 \ &=& 2p - 1\ &=& p - (1-p) \end{eqnarray} $$