Questions: Variance and Standard Deviation of Random Variables

Question 1

A random variable X has variance 9. You define Y = 3X + 7. What is Var(Y)?

• A. 27
• B. 81
• C. 34
• D. 9

Answer: (B)

Var(aX + b) = a²·Var(X). Here a = 3 and b = 7, so Var(Y) = 3²·9 = 9·9 = 81. The constant +7 shifts every outcome equally without changing spread, so it contributes nothing. A common error is multiplying by a instead of a² (giving 27), or adding 7 to the result (giving 34).

Question 2

If X and Y are independent random variables with standard deviations 3 and 4 respectively, then the standard deviation of X + Y is 7.

• T. True
• F. False

Answer: False

Standard deviations do not add — variances do (for independent variables). Var(X + Y) = Var(X) + Var(Y) = 9 + 16 = 25, so SD(X + Y) = √25 = 5, not 3 + 4 = 7. This is the 3-4-5 right triangle appearing in probability: standard deviations combine like sides of a right triangle, not like collinear lengths.

Question 3

Why is variance defined using squared deviations E[(X − μ)²] rather than the simpler expected absolute deviation E[|X − μ|]?

Model answer: Squared deviations are algebraically tractable: they yield the clean rules Var(aX + b) = a²Var(X) and Var(X + Y) = Var(X) + Var(Y) for independent variables. Absolute value is not differentiable at zero and does not obey these addition rules.

E[|X − μ|] (mean absolute deviation) is more interpretable in isolation but breaks down algebraically. The squaring operation is what gives variance its composition rules — the ones that power almost all of statistical theory. Standard deviation σ = √Var(X) then restores interpretable units while retaining the algebraic tractability inherited from variance.