If X is a random variable, what would the VARIANCE of Y be if Y=X+2
If Y = X + 2, then the variance of Y can be calculated as follows:
Var(Y) = Var(X + 2)
Since Var(aX) = a^2Var(X), for any constant 'a', we can rewrite the above equation as:
Var(Y) = Var(X)
Therefore, the variance of Y would be equal to the variance of X.
To find the variance of Y, we need to know two things - the variance of X and the covariance between X and the constant 2.
1. Variance of X:
The variance of X, denoted as Var(X), is a measure of how much the values of X vary or spread out around the mean. It can be calculated as the average of the squared differences from the mean value of X.
2. Covariance between X and the constant 2:
Since Y is defined as Y = X + 2, we need to consider the covariance between X and the constant 2. The covariance, denoted as Cov(X, Y), measures the relationship between the two variables and indicates how they change together.
The variance of Y can be calculated as follows:
Var(Y) = Var(X + 2) = Var(X) + Var(2) + 2 * Cov(X, 2)
However, since 2 is a constant, the variance of a constant is zero (Var(2) = 0). Additionally, the covariance between X and a constant is also zero (Cov(X, 2) = 0). Therefore, these terms can be omitted, simplifying the equation to:
Var(Y) = Var(X)
In conclusion, the variance of Y with Y = X + 2 is equal to the variance of X.
To find the variance of Y, we can use the properties of variance. Here are the steps:
Step 1: Begin with the definition of variance. The variance of a random variable Y is defined as Var(Y) = E[(Y - E(Y))^2], where E(Y) represents the expected value of Y.
Step 2: Substitute the expression for Y into the equation. We have Var(Y) = E[((X + 2) - E(X + 2))^2].
Step 3: Expand the expression. This gives us Var(Y) = E[(X + 2)^2 - 2*(X + 2)*E(X + 2) + [E(X + 2)]^2].
Step 4: Simplify the expression. Since E(X + 2) is a constant, we can rewrite E[(X + 2)^2] as [E(X + 2)]^2, making the second term in the expression equal to zero.
Step 5: Now we are left with Var(Y) = E[(X + 2)^2] - [E(X + 2)]^2.
Step 6: Simplify further. Expand (X + 2)^2 to get Var(Y) = E[X^2 + 4X + 4] - [E(X + 2)]^2.
Step 7: Using the linearity property of expectation, we can split E[E(X^2 + 4X + 4)] as E(X^2) + E(4X) + E(4).
Step 8: Since X is a random variable, it will contribute to the variance, so Var(Y) = Var(X) + 4*Var(X) + 0.
Step 9: Combine like terms. Var(Y) = 5*Var(X).
Therefore, the variance of Y is 5 times the variance of X.