Question

Suppose α, β, γ denote the proportions of Democrats (D), Republicans (R) and Others (0) in a large population of voters. (So 0-α, β, γ 1 and α+β+7 1.) An individual is selected at random from the population. Write x-, y 0, Z-0 if that individual is D, write X = 0, Y = 1, Z 0 if the individual is R and write X-0, Y = 0, Z if the individual is O. Find: E(X), E(Y)

Answer 1

The expected value of X (Republicans) is β, and the expected value of Y (Democrats) is α.

The expected value (E) of a random variable represents the average value of that variable in the long run. In this case, we're interested in finding the expected values of X (Republicans) and Y (Democrats).

Given α represents the proportion of Democrats, β represents the proportion of Republicans, and γ represents the proportion of Others, we can express the probabilities of selecting each group:

P(X = 1) = β (proportion of Republicans)

P(Y = 1) = α (proportion of Democrats)

The expected value of a random variable can be calculated as follows:

E(X) = ∑(x * P(X = x)) for all possible values of X

E(Y) = ∑(y * P(Y = y)) for all possible values of Y

So, for X (Republicans):

E(X) = 1 * P(X = 1) + 0 * P(X = 0)

E(X) = 1 * β + 0 * (1 - β)

E(X) = β

And for Y (Democrats):

E(Y) = 1 * P(Y = 1) + 0 * P(Y = 0)

E(Y) = 1 * α + 0 * (1 - α)

E(Y) = α

Therefore, the expected value of X (Republicans) is β, and the expected value of Y (Democrats) is α.