Question: Consider a U.S. Senate election in a southern state. Let

d = P{D} = probability of an arbitrary voter voting for D
r = P{R} = probability of an arbitrary voter voting for R

It turned out that 2,000,000 people voted. In an exit poll, conducted involving 20,000 voters, candidates D and R received 10,020 and 9,980 votes, respectively. Define a random variable for an arbitrary voter

\[{{X}_{i}}=\left\{ \begin{aligned} & 1\text{ }\,\,\,\,\text{if the voter votes for D} \\ & -1\text{ if the voter votes for R} \\ \end{aligned} \right.\]

and let Y = (total votes for D) - (total votes for R).

(a) Assuming a vote is independent of any other vote, determine the expected values and variances of X and Y, respectively, in terms of exact d and r
(b) Based on the exit poll, use the sample means d(bar) and r(bar) of the probabilities d and r, respectively, to determine which candidate (D or R) will win and with how much probability.
(c) Based on the exit poll, predict how many more votes the winner (D or R) will have than the loser (R or D).
(d) Based on the exit poll, determine (approximately) the standard deviation of Y. Give concrete values of the standard deviation.