SB3-SB4 Mean, variance and sums of Poisson distributions Quiz: AQA Further Maths, Unit 4

The 20 questions

1. For X ~ Po(lambda), what is E(X)?

   • square root of lambda
   • lambda squared
   • lambda
   • 1/lambda

2. For X ~ Po(lambda), what is Var(X)?

   • 1/lambda
   • 2 lambda
   • lambda
   • lambda squared

3. For X ~ Po(lambda), what is the standard deviation of X?

   • lambda squared
   • sqrt(lambda)
   • 1 / sqrt(lambda)
   • lambda

4. If X ~ Po(lambda) and Y ~ Po(mu) are independent, what is the distribution of X + Y?

   • Poisson with parameter lambda + mu
   • Poisson with parameter lambda mu
   • Binomial with n = lambda + mu
   • Uniform on the interval from 0 to lambda + mu

5. Which statement about the difference X - Y of independent Poisson variables X and Y is correct?

   • It is Poisson with mean lambda times mu
   • It is not Poisson because it can take negative values
   • It is always equal to zero
   • It is Poisson with parameter lambda minus mu

6. Which property of data suggests a Poisson model may be appropriate?

   • The mean and variance are approximately equal
   • The data are all whole numbers between 0 and 1
   • The mean is much larger than the variance
   • The data are symmetric about zero

7. If X ~ Po(lambda) and Y ~ Po(mu) are independent, what is Var(X + Y)?

   • lambda mu
   • (lambda + mu)^2
   • lambda + mu
   • sqrt(lambda + mu)

8. X ~ Po(3) and Y ~ Po(4) are independent. What is P(X + Y = 0) to 6 decimal places in terms of e?

   • e^(-7), about 0.000912
   • e^(-4), about 0.0183
   • e^(-12), about 0.00000614
   • e^(-3), about 0.0498

9. X ~ Po(2) and Y ~ Po(5) are independent. What is Var(X + Y)?

   • 10
   • 2.5
   • sqrt(7)
   • 7

10. X ~ Po(16). What is the standard deviation of X?

    • 8
    • 256
    • 16
    • 4

11. X ~ Po(3) and Y ~ Po(5) are independent. What is E(X + Y)?

    • 15
    • 2
    • 8
    • 3.5

12. X ~ Po(4) per hour and Y ~ Po(2) per hour are independent. What is P(X + Y = 0) for one hour, to 4 decimal places?

    • 0.1353
    • 0.0498
    • 0.0183
    • 0.0025

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

    • 12
    • 3
    • 6
    • 9

14. X ~ Po(4) and Y ~ Po(6) are independent. What is the standard deviation of X + Y?

    • 10
    • sqrt(6), about 2.45
    • sqrt(10), about 3.16
    • 4

15. Over three hours the total number of events is Po(9). If each hour is Po(lambda) and the hours are independent, what is lambda?

    • 1/3
    • 3
    • 27
    • 9

16. For X ~ Po(lambda), what is E(X^2) in terms of lambda?

    • lambda + lambda squared
    • lambda squared
    • 2 lambda
    • lambda

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

    • 12
    • 9
    • 3
    • 6

18. X ~ Po(2) and Y ~ Po(3) are independent. What is P(X + Y = 1) in terms of e, to 4 decimal places?

    • 5e^(-5), about 0.0337
    • e^(-5)/5, about 0.0013
    • 3e^(-3), about 0.1494
    • 2e^(-2), about 0.2707

19. X ~ Po(4) and Y ~ Po(1) are independent. What is P(X + Y >= 2) to 4 decimal places?

    • 0.0337
    • 0.9500
    • 0.9596
    • 0.0404

20. X1, X2 and X3 are independent, each Po(2). What is P(X1 + X2 + X3 = 0)?

    • e^(-6)
    • 3e^(-2)
    • e^(-8)
    • e^(-2)

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