3B.7.1 Definition and derivation of probability generating functions Quiz: Pearson Edexcel Further Maths, Unit 16

The 20 questions

1. What is the definition of the probability generating function G_X(t) of a discrete random variable X?

   • G_X(t) = E(X^t)
   • G_X(t) = E(e^(tX))
   • G_X(t) = E(t^X)
   • G_X(t) = E(tX)

2. What is the value of G_X(1) for any discrete random variable?

   • t
   • 1
   • E(X)
   • 0

3. What is the probability generating function of a Poisson distribution with parameter lambda?

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

4. What is the probability generating function of a binomial distribution B(n, p)?

   • np t^n
   • (p + (1 - p)t)^n
   • (1 - p + pt)^n
   • 1 - p + p t^n

5. What is the probability generating function of a geometric distribution with parameter p?

   • pt / (1 - pt)
   • p / (1 - (1 - p)t)
   • pt / (1 - (1 - p)t)
   • (1 - p)t / (1 - pt)

6. What is the probability generating function of a fair six-sided die?

   • 6t/(1 - t)
   • t^6/6
   • (1 + t + t^2 + t^3 + t^4 + t^5)/6
   • (t + t^2 + t^3 + t^4 + t^5 + t^6)/6

7. A Bernoulli random variable has P(X = 1) = 0.3. What is its PGF?

   • 0.7t + 0.3t^2
   • 0.3t
   • 0.3 + 0.7t
   • 0.7 + 0.3t

8. X ~ Po(2). What is G_X(0)?

   • 0
   • 1
   • e^(-2)
   • 2

9. A random variable has G(t) = 0.2 + 0.5t + 0.3t^2. What is P(X = 1)?

   • 0.8
   • 0.3
   • 0.5
   • 0.2

10. A random variable has G(t) = 0.1t + 0.4t^2 + 0.5t^3. What is P(X = 2)?

    • 0.9
    • 0.5
    • 0.1
    • 0.4

11. For X ~ B(2, p) with q = 1 - p, what is the coefficient of t in the PGF?

    • p^2
    • 2p
    • q^2
    • 2pq

12. For a geometric distribution with p = 0.5, what is G(0.5)?

    • 1/2
    • 2/3
    • 1/4
    • 1/3

13. What is the coefficient of t^k in the probability generating function G_X(t)?

    • P(X = k)
    • E(X = k)
    • k P(X = k)
    • P(X >= k)

14. What is the PGF of the sum X + Y of independent random variables?

    • G_X(t) - G_Y(t)
    • G_X(t^2)
    • G_X(t) x G_Y(t)
    • G_X(t) + G_Y(t)

15. X ~ Po(1) and Y ~ Po(2) are independent. What is G_{X+Y}(t)?

    • e^(2(t - 1))
    • e^(2t - 3)
    • e^(3(t - 1))
    • e^(3t)

16. A random variable has G(t) = (0.6 + 0.4t)^5. What is P(X = 0)?

    • 0.01024
    • 0.6
    • 0.4
    • 0.07776

17. A random variable has G(t) = k t(1 + t) with G(1) = 1. What is k?

    • 1/4
    • 1
    • 1/2
    • 2

18. A random variable has G(t) = 0.25 + 0.75t^3. What is P(X = 0)?

    • 1
    • 0.75
    • 0.25
    • 0

19. Which property means a probability generating function determines the distribution uniquely?

    • The PGF equals the mean of the distribution, which is a single number
    • The PGF is always a polynomial in t of finite degree for every variable
    • The PGF determines all the probabilities of the distribution
    • The PGF is only defined when t = 1, so it can only give the total probability

20. X and Y are independent Bernoulli random variables with P(X = 1) = P(Y = 1) = p. What is the PGF of X + Y?

    • (1 - p + p t^2)
    • (1 - p + pt)^2
    • (1 - p + pt)
    • (1 - p)^2 + p^2 t

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