3B.7.2-3B.7.3 Using probability generating functions for mean and variance Quiz: Pearson Edexcel Further Maths, Unit 16

The 20 questions

1. In terms of the PGF G(t), what is E(X)?

   • G'(1)
   • G''(1)
   • G'(0)
   • G(1)

2. In terms of the PGF G(t), what is Var(X)?

   • G''(1) - G'(1) - [G'(1)]^2
   • G''(0) - G'(0) + [G'(0)]^2
   • G'(1) + [G'(1)]^2 - G''(1)
   • G''(1) + G'(1) - [G'(1)]^2

3. For X ~ Po(lambda), what is G'(1)?

   • lambda
   • 1
   • lambda^2
   • 0

4. For X ~ Po(lambda), what is G''(1)?

   • 0
   • e^lambda
   • lambda
   • lambda^2

5. What is the mean of a geometric distribution with p = 0.2 found from its PGF?

   • 5
   • 4
   • 0.8
   • 0.2

6. A random variable has G(t) = (t + t^2)/2. What is G'(1)?

   • 1.5
   • 2
   • 0.5
   • 1

7. For X ~ B(n, p), what is G'(1)?

   • p/n
   • n(1 - p)
   • np
   • n^2 p

8. Given G(t) = (0.4 + 0.6t)^3, what is Var(X)?

   • 0.24
   • 1.8
   • 0.72
   • 0.6

9. X and Y are independent with X ~ Po(2) and Y ~ Po(3). What is E(X + Y)?

   • 2.5
   • 1
   • 6
   • 5

10. X and Y are independent with X ~ Po(2) and Y ~ Po(3). What is Var(X + Y)?

    • 5
    • 13
    • 9
    • 2

11. For G(t) = 0.2 + 0.3t + 0.5t^2, what is G'(1)?

    • 1
    • 1.3
    • 0.8
    • 1.5

12. X ~ B(2, 0.5) and Y ~ B(3, 0.5) are independent. What is E(X + Y)?

    • 2.5
    • 2
    • 1.5
    • 3

13. If two PGFs are equal, what can be concluded about the random variables?

    • They are independent
    • They have the same variance only
    • They have the same mean only
    • They have the same distribution

14. A random variable has G(t) = e^(4(t - 1)). What is P(X = 0)?

    • 4e^(-4)
    • 1 - e^(-4)
    • e^4
    • e^(-4)

15. G_X(t) = (0.5 + 0.5t)^2. What is E(X)?

    • 0.25
    • 1
    • 0.5
    • 2

16. What does E(X(X - 1)) equal in terms of the PGF?

    • G''(1)
    • E(X)
    • Var(X)
    • E(X^2)

17. G(t) = (t + t^2 + t^3)/3. What is E(X)?

    • 6
    • 3
    • 2
    • 1

18. Three independent Po(1) variables are summed. What is the PGF of the sum?

    • e^(3t - 1)
    • e^(t - 3)
    • e^(3(t - 1))
    • 3e^(t - 1)

19. Why is the PGF useful when working with sums of independent random variables?

    • The PGF of the product is the sum of the PGFs
    • The PGF of the sum is the sum of the individual PGFs
    • The PGF of the sum equals the derivative of each PGF
    • The PGF of the sum is the product of the individual PGFs

20. X has PGF G(t) = (0.3 + 0.7t)^4. What is E(X)?

    • 4
    • 2.8
    • 0.84
    • 1.2

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