4B.4.1 Distributions of linear combinations of normal variables Quiz: Pearson Edexcel Further Maths, Unit 21

The 20 questions

1. If X is normal with mean mu and variance sigma^2, what is the variance of aX + b?

   • a sigma + b
   • a^2 sigma^2
   • a^2 sigma + b
   • a sigma^2 + b

2. What is the variance of the sum of two independent normal random variables with variances V1 and V2?

   • V1 + V2
   • V1 V2
   • V1^2 + V2^2
   • V1 - V2

3. If X and Y are independent with variances sigma_x^2 and sigma_y^2, what is Var(X - Y)?

   • sigma_x^2 - sigma_y^2
   • (sigma_x - sigma_y)^2
   • sigma_x + sigma_y
   • sigma_x^2 + sigma_y^2

4. What property must independent normal variables have for their variances to be additive?

   • Equal variance
   • Standard normal
   • Zero mean
   • Zero covariance

5. If X ~ N(5, 2) and Y ~ N(3, 4) are independent, with the second parameter a variance, what is the distribution of X - Y?

   • N(2, 6)
   • N(2, 4)
   • N(2, 2)
   • N(8, 6)

6. If E(X) = 4 and E(Y) = 1, what is E(3X - 2Y)?

   • 8
   • 10
   • 14
   • 11

7. In the notation X ~ N(mu, sigma^2), what does the second parameter sigma^2 represent?

   • Mean
   • Variance
   • Standard deviation
   • Median

8. X ~ N(10, 4) and Y ~ N(5, 9) are independent. What is the distribution of 2X - Y?

   • N(5, 25)
   • N(15, 25)
   • N(15, 7)
   • N(15, 13)

9. X ~ N(10, 4) and Y ~ N(5, 9) are independent. What is P(2X - Y > 15)?

   • 0.1587
   • 0.3413
   • 0.8413
   • 0.5

10. X ~ N(10, 4) and Y ~ N(5, 9) are independent. What is the standard deviation of 3X + Y, to 2 decimal places?

    • 6.71
    • 6.00
    • 4.12
    • 45.00

11. X ~ N(20, 9) and Y ~ N(12, 16) are independent. What is the distribution of X - Y?

    • N(8, 7)
    • N(32, 25)
    • N(8, 1)
    • N(8, 25)

12. X and Y are independent, with E(X) = 2, E(Y) = -1. What is E(4X + 3Y)?

    • 11
    • 5
    • 3
    • -5

13. X ~ N(1, 1) and Y ~ N(3, 4) are independent. What is the standard deviation of 2X + Y?

    • 8.00
    • 2 sqrt(2), about 2.83
    • 3.46
    • 4.00

14. X ~ N(10, 4) and Y ~ N(5, 9) are independent. What is P(X - Y > 7), to 2 decimal places?

    • 0.71
    • 0.58
    • 0.42
    • 0.29

15. Three independent variables each follow N(4, 2). What is the distribution of their sum?

    • N(4, 6)
    • N(12, 2)
    • N(12, 18)
    • N(12, 6)

16. X ~ N(50, 16) and Y ~ N(40, 9) are independent. What is P(X > Y), to 3 decimal places?

    • 0.977
    • 0.500
    • 0.023
    • 0.841

17. X ~ N(2, 4). If Y = aX + b has Y ~ N(7, 36) with a > 0, which pair (a, b) is correct?

    • a = 3, b = 7
    • a = 6, b = -5
    • a = 3, b = 1
    • a = -3, b = 13

18. If X and Y are independent, what is the correct formula for Var(X - Y) in terms of their variances?

    • Var(X) - Var(Y)
    • Var(X) + Var(Y)
    • Var(X) * Var(Y)
    • Var(X) / Var(Y)

19. Four independent bags each have weight N(500, 25) grams. What is P(total weight > 2010), to 4 decimal places?

    • 0.8413
    • 0.0500
    • 0.3173
    • 0.1587

20. X and Y are independent, each N(0, 1). What is the distribution of (X + Y)/sqrt(2)?

    • N(0, 1/2)
    • N(0, 1)
    • N(0, sqrt(2))
    • N(0, 2)

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