Self-Assessment Quiz: Understanding Var(aX + b) and Standard Deviation Rules

📚 Quick Study Guide

• 🔢 Variance of a Constant: The variance of a constant is always zero. $Var(c) = 0$
• ➕ Effect of Adding a Constant: Adding a constant to a variable doesn't change its variance. $Var(X + b) = Var(X)$
• ✖️ Effect of Multiplying by a Constant: Multiplying a variable by a constant scales the variance by the square of the constant. $Var(aX) = a^2Var(X)$
• ➗ Combined Transformation: For a linear transformation $aX + b$, the variance is $Var(aX + b) = a^2Var(X)$
• 📏 Standard Deviation: The standard deviation is the square root of the variance. $SD(X) = \sqrt{Var(X)}$
• ✅ Standard Deviation of a Constant: $SD(c) = 0$
• 💡 Standard Deviation Transformation: $SD(aX + b) = |a|SD(X)$

Practice Quiz

1. What is the variance of the constant 5?
   1. A) 0
   2. B) 5
   3. C) 25
   4. D) Cannot be determined
2. If $Var(X) = 9$, what is $Var(X + 3)$?
   1. A) 3
   2. B) 6
   3. C) 9
   4. D) 12
3. If $Var(X) = 4$, what is $Var(2X)$?
   1. A) 2
   2. B) 4
   3. C) 8
   4. D) 16
4. If $Var(X) = 16$, what is $Var(3X + 2)$?
   1. A) 14
   2. B) 48
   3. C) 144
   4. D) 146
5. If $SD(X) = 5$, what is $SD(2X + 1)$?
   1. A) 5
   2. B) 6
   3. C) 10
   4. D) 11
6. If $SD(X) = 3$, what is $SD(-X + 4)$?
   1. A) -3
   2. B) 1
   3. C) 3
   4. D) 7
7. Given $Var(X) = 25$, what is $SD(4X - 3)$?
   1. A) 7
   2. B) 17
   3. C) 20
   4. D) 100