Question

Given two random variables X and Y, the expected value of their sum, E(X+Y), is equal to the sum of their individual expected values, E(X) and E(Y), minuses their correlation, which can be positive or negative. True False

Answer 1

Explanation:

False.

The expected value of the sum of two random variables X and Y is always equal to the sum of their individual expected values, E(X) and E(Y), regardless of their correlation. That is, E(X + Y) = E(X) + E(Y).

The covariance of X and Y, which is a measure of their linear association or correlation, is given by Cov(X, Y) = E[(X - E(X))(Y - E(Y))]. The correlation between X and Y, denoted by ρ(X, Y), is defined as the covariance between X and Y divided by the product of their standard deviations, that is, ρ(X, Y) = Cov(X, Y) / (σ(X)σ(Y)).

The correlation between X and Y can affect their joint distribution and the probability of certain outcomes, but it does not affect the expected value of their sum. Therefore, the statement that E(X + Y) is equal to the sum of their individual expected values minus their correlation is false.