Question

There are N voters. There are 4 candidates. If we were to ask each of the N voters, we would learn that

30% would vote for candidate A

25% would vote for candidate B

25% would vote for candidate C

20% would vote for candidate D

Since it is unrealistic to ask the whole population, we randomly sample a subset n. Let Yi be the response from the i th sampled voter. Let yi denote the outcome of Yi.

yi = 1 if voter votes for A

= 2 if voter votes for B

= 3 voter votes for C

= 4 voter votes for D.

(a) What's the probability of each outcome Yi? P(Yi=1), P(Yi=2), P(Yi=3), P(Yi=4)?

(b) What's the E(Yi)? Can we find P(Yi=yi) from E(Yi)?

(c) What's the estimator of the proportion of voters who support candidate C?

Answer 1

a)

P(Yi=1) = 0.3 ( i.e, Probability that voters votes for A)

P(Yi=2) = 0.25 ( i.e, Probability that voters votes for B)

P(Yi=3)= 0.25 ( i.e, Probability that voters votes for C)

P(Yi=4) = 0.2   ( i.e, Probability that voters votes for D)

b) E(Yi)=   = 1 * 0.3 + 2*0.25 + 3*0.25 + 4 * 0.2

= 2.35

No , we cannot   find P(Yi=yi) from E(Yi)

c) estimator of the proportion of voters who support candidate C = 0.2