Homework 19: Indicator Random Variables

1. You randomly throw 6 balls into 10 different baskets. Let \(X\) be the number of balls which land in the first basket, and let \(Y\) be the number of baskets which are empty.
   1. Find \(f_X(x)\).
   2. Find \(E(X)\).
   3. Find \(E(Y)\). (Hint. Let \(I_j\) be the indicator of the event ``basket \(j\) is empty''. Note that \(Y\) is the sum of the \(I_j\)'s.)
2. From a group of 8 math majors and 7 physics majors (no double majors), 4 are randomly selected for the Putnam Competition team. Let \(M\) be the number of math majors on the team. Find \(E(M)\) in two ways:
   1. Use the definition of \(E(M)\). (You will need to first find \(f_M(m)\).)
   2. Use indicator random variables. Specifically, let \(I_j\) be the indicator of the event ``the \(j\)-th person selected is a math major''. Use the fact that \(M=I_1+I_2+I_3+I_4\).
3. Ten husband and wife couples are randomly seated in a circle. Find the expected number of husbands who are seated next to their wives.
4. Suppose a bent coin has a probability \(p=0.4\) of landing heads. If one flips the coin \(n\) times, what is the expected number of heads which are immediately followed by a tail. (For example, if \(n=8\), then for outcome 'THHTTHTH', two heads, namely the third and sixth flips, are immediately followed by a tail.)
5. A population of \(n\) people vote in an election. \(d\) vote democratic and \(n-d\) vote republican. In the next election, the probability of a democratic voter switching to republican is \(p_1\), and the probability of a republican voter switching to democratic is \(p_2\). Let \(X\) be the number of democratic votes in the second election. Find \(E(X)\).