Question

In a certain town, 40% of the eligible voters prefer candidate a, 10% prefer candidate b, and the remaining 50% have no preference. you randomly sample 12 eligible voters.

a) what is the probability that 6 will prefer candidate a, 2 will prefer candidate b, and the remaining 4 will have no preference?
b) what is the marginal distribution for the number of voters who prefer candidate a in your sample?
c) what is the marginal joint pmf for the number of voters who prefer candidates a, b in your sample?
d) what is the conditional joint distribution for the number of voters who prefer candidates a, b in your sample given that 5 voters in the sample have no preference?

Answer 1

The distribution of X, the number of eligible voters who prefer candidate a in a sample of 12, can be approximated by a binomial distribution. The marginal distribution for the number of voters who prefer candidate a in the sample can be calculated using the binomial probability formula.

Step-by-step explanation:

a) The distribution of X, the number of eligible voters who prefer candidate a in a sample of 12, can be approximated by a binomial distribution with parameters n=12 and p=0.4, where n is the sample size and p is the probability of success (i.e., the probability that an eligible voter prefers candidate a).

b) To find the marginal distribution for the number of voters who prefer candidate a in the sample, we can use the binomial probability formula for each possible outcome.

The probabilities for each outcome can be calculated using the binomial probability formula: P(X=k) = nCk * p^k * (1-p)^(n-k), where nCk is the number of combinations of n things taken k at a time.

c) The marginal joint probability mass function (pmf) for the number of voters who prefer candidates a and b in the sample can be calculated by multiplying the individual probabilities of each outcome. For example, P(X=6 and Y=2) = P(X=6) * P(Y=2) = P(X=6) * P(X=2) = (nC6 * p^6 * (1-p)^(n-6)) * (nC2 * p^2 * (1-p)^(n-2)).

d) The conditional joint distribution for the number of voters who prefer candidates a and b in the sample, given that 5 voters have no preference, can be found by considering only the remaining 7 voters. We can calculate the conditional joint probabilities using the same method as in part c.