Lesson 4B.4.1

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

20 questions

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Lesson 4B.4.1, Distributions of linear combinations of normal variables: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 21: Linear combinations of normal random variables, written with Revision Ninja.

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The 20 questions

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

    • Normal with mean a mu and variance sigma^2 + b
    • Normal with mean a mu + b and variance a^2 sigma^2
    • Normal with mean mu and variance a^2 sigma^2 + b^2
    • Normal with mean a mu + b and variance a sigma^2

  2. What is the distribution of the sum of two independent normal random variables?

    • Normal, with mean equal to the sum of the means and variance equal to the sum of the variances
    • Normal, with standard deviation equal to the sum of the standard deviations
    • Normal, with the same mean and variance equal to the sum of the variances
    • Uniform, with range equal to the sum of the ranges

  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. Why is independence needed for the variance formula of a linear combination of normal variables?

    • So that the coefficients a and b are both positive
    • So that the total variance of the combination is zero
    • So that the means of the two variables are equal
    • So that the covariance is zero, which lets the variances add

  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. What does X ~ N(mu, sigma^2) mean?

    • X is normal with mean sigma^2 and variance mu
    • X is normal with mean mu and variance sigma^2
    • X is normal with mean mu and standard deviation sigma^2
    • X is normal with mean mu and variance 1/sigma^2

  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. A student says that if X and Y are independent normal variables, then Var(X - Y) = Var(X) - Var(Y). What is the error?

    • Variances are subtracted when the mean is subtracted
    • Variances always add for independent variables, so Var(X - Y) = Var(X) + Var(Y)
    • Variances are subtracted when the variables are negatively correlated
    • The variance of a difference of normal variables does not exist

  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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