If X and Y are two variables, both with a variance equal to 1 and a covariance equal to 0.5, we need to find the variance of the expression 2X + Y + 1.
To find the variance of a linear combination of variables, we can use the following formula:
Var(aX + bY + c) = a^2 * Var(X) + b^2 * Var(Y) + 2ab * Cov(X, Y)
In this case, a = 2, b = 1, c = 1, Var(X) = 1, Var(Y) = 1, and Cov(X, Y) = 0.5.
Plugging these values into the formula, we get:
Var(2X + Y + 1) = (2^2 * 1) + (1^2 * 1) + 2 * 2 * 1 * 0.5
Simplifying further:
Var(2X + Y + 1) = 4 + 1 + 2
Var(2X + Y + 1) = 7
Therefore, the variance of 2X + Y + 1 is equal to 7.