SF3-SF4 Mean and variance, and the link to Poisson events Quiz: AQA Further Maths, Unit 4

The 20 questions

1. For an exponential distribution with rate lambda, what is E(X)?

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

2. For an exponential distribution with rate lambda, what is Var(X)?

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

3. For an exponential distribution with rate lambda, what is the standard deviation of X?

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

4. For an exponential distribution, which statement about the mean and standard deviation is correct?

   • The mean is larger than the standard deviation
   • They are equal
   • The mean is the square of the standard deviation
   • The standard deviation is twice the mean

5. The lengths of the intervals between events in a Poisson process with rate lambda follow which distribution?

   • Uniform on [0, lambda]
   • Exponential with rate lambda
   • Poisson with mean 1/lambda
   • Normal with mean lambda

6. What integral gives the mean of the exponential distribution with pdf lambda e^(-lambda x) for x >= 0?

   • Integral of x lambda e^(-lambda x) from 0 to infinity
   • Integral of e^(-lambda x) from 0 to infinity
   • Integral of lambda e^(-lambda x) from 0 to infinity
   • Integral of x squared lambda e^(-lambda x) from 0 to infinity

7. For a Poisson process with rate lambda, what is the parameter of the exponential distribution of the interval between events?

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

8. Events occur at 4 per hour, modelled as a Poisson process. What is the mean time between events?

   • 15 minutes
   • 60 minutes
   • 4 minutes
   • 0.4 hours

9. For an exponential distribution with rate lambda = 0.5, what is the mean?

   • 0.25
   • 2
   • 4
   • 0.5

10. For an exponential distribution with rate lambda = 0.5, what is the variance?

    • 4
    • 0.5
    • 0.25
    • 2

11. Events occur at a rate of 3 per minute, modelled as a Poisson process. What is P(the first interval is less than 20 seconds), to 4 decimal places?

    • 0.9502
    • 0.3679
    • 0.6321
    • 0.0498

12. Events occur at 6 per hour, modelled as a Poisson process. What is the probability of no event in 10 minutes, to 4 decimal places?

    • 0.3679
    • 0.9502
    • 0.6321
    • 0.0498

13. For an exponential distribution with rate lambda = 0.8, what is the mean?

    • 1.25
    • 0.8
    • 1.6
    • 0.64

14. For an exponential distribution with rate lambda = 5, what is the standard deviation?

    • 0.04
    • 0.2
    • 5
    • 25

15. Events occur at a rate of 2 per minute, modelled as a Poisson process. What is the probability that an interval between events exceeds 1 minute, to 4 decimal places?

    • 0.3679
    • 0.8647
    • 0.1353
    • 0.6321

16. Events occur as a Poisson process at rate 4 per hour. What is the probability that the next event occurs within 30 minutes, to 4 decimal places?

    • 0.6321
    • 0.1353
    • 0.8647
    • 0.3679

17. In the derivation of the mean of an exponential distribution, which technique is typically used to evaluate the integral of x lambda e^(-lambda x)?

    • Substitution of x = 0 only
    • Use of the normal approximation
    • Integration by parts
    • Differentiation of the cdf

18. For an exponential distribution, what is E(X^2)?

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

19. For an exponential distribution with rate lambda, what is the value of lambda if the mean is 5 minutes?

    • 0.05
    • 0.2
    • 0.5
    • 5

20. An exponential distribution has rate lambda = 0.5 per minute. What is P(X > mean), to 4 decimal places?

    • 0.5000
    • 0.3679
    • 0.1353
    • 0.6321

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