There are 1000 households in a town. Specifically, there are 100 households with one member, 200 households with 2 members, 300 households with 3 members, 200 households with 4 members, 100 households with 5 members, and 100 households with 6 members. Thus, the total number of people living in the town is N = 100 ⋅ 1 +200 ⋅ 2 +300 ⋅ 3 +200 ⋅ 4 +100 ⋅ 5 +100 ⋅ 6 = 3300.

a. We pick a household at random, and define the random variable X as the number of people in the chosen household. Find the PMF and the expected value of X.

b. We pick a person in the town at random, and define the random variable Y as the number of people in the household where the chosen person lives. Find the PMF and the expected value of Y .