3B.2.2-3B.2.3 Mean, variance and the Poisson approximation to the binomial Quiz: Pearson Edexcel Further Maths, Unit 11

The 20 questions

1. For X ~ B(n, p), what is E(X)?

   • n(1 - p)
   • p/n
   • np
   • n + p

2. For X ~ B(n, p), what is Var(X)?

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

3. For Y ~ Po(lambda), what is Var(Y)?

   • 1/lambda
   • lambda
   • lambda^2
   • 2 lambda

4. When is the Poisson distribution a good approximation to B(n, p)?

   • When n is small and p is large
   • When n is small and p is close to 1
   • When n is large and p is small
   • When p is always 0.5

5. What Poisson parameter is used to approximate B(n, p)?

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

6. For X ~ B(20, 0.1), what are the mean and variance?

   • Mean 0.1, variance 1.8
   • Mean 2, variance 2
   • Mean 18, variance 1.8
   • Mean 2, variance 1.8

7. X ~ B(100, 0.02) is approximated by Po(2). What is P(X = 0) to 4 decimal places?

   • 0.0183
   • 0.8647
   • 0.2707
   • 0.1353

8. Y ~ B(50, 0.04) is approximated by Po(2). What is P(Y = 2) to 4 decimal places?

   • 0.2240
   • 0.1353
   • 0.2707
   • 0.5413

9. Under the Poisson approximation to B(100, 0.02), what is the variance?

   • 1.96
   • 0.02
   • 2
   • 4

10. A Poisson distribution has mean 6. What is its variance?

    • 6
    • 36
    • 3
    • sqrt(6)

11. X ~ B(n, p) has E(X) = 12 and Var(X) = 8. What is p?

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

12. X ~ B(1000, 0.001) is approximated by Po(1). What is P(X >= 2) to 4 decimal places?

    • 0.1839
    • 0.3679
    • 0.7358
    • 0.2642

13. Y = 3X where X ~ Po(2). What is Var(Y)?

    • 6
    • 2
    • 18
    • 12

14. X ~ B(n, p) has mean 5 and variance 4. What is n?

    • 20
    • 50
    • 10
    • 25

15. A process has a 2% defect rate and 150 items are inspected. Using a Poisson approximation, what is the probability of no defects to 4 decimal places?

    • 0.1494
    • 0.2240
    • 0.0183
    • 0.0498

16. X ~ B(10, 0.3). What is E(X^2)?

    • 9
    • 3
    • 11.1
    • 2.1

17. If X ~ Po(3), what is E(X^2)?

    • 9
    • 15
    • 12
    • 3

18. Y ~ B(40, 0.0125) is approximated by Po(0.5). What is P(Y <= 1) to 4 decimal places?

    • 0.3033
    • 0.6065
    • 0.9098
    • 0.1494

19. Which statement about the Poisson approximation to the binomial is correct?

    • It is more accurate when n increases and p decreases with np fixed
    • It is exact for every value of n and p, with no approximation error at all
    • It is most accurate when p is close to 1 and n is small in size
    • It is worse as n increases with p fixed, because the variance grows

20. For X ~ B(500, 0.01), what is the exact variance?

    • 4.9
    • 5.05
    • 5
    • 4.95

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