SF1-SF2 Modelling and probabilities with the exponential distribution Quiz: AQA Further Maths, Unit 4

The 20 questions

1. What is the probability density function of an exponential distribution with rate lambda?

   • f(x) = lambda x for x >= 0
   • f(x) = 1/lambda for x >= 0
   • f(x) = lambda e^(-lambda x) for x >= 0
   • f(x) = e^(-lambda x) for 0 <= x <= 1

2. What is the cumulative distribution function of an exponential distribution with rate lambda?

   • F(x) = 1 - lambda x
   • F(x) = 1 - e^(-lambda x)
   • F(x) = lambda e^(-lambda x)
   • F(x) = e^(-lambda x)

3. Which conditions make an exponential distribution a suitable model for a situation?

   • Events occur in fixed groups at fixed times
   • Each event has exactly the same size
   • Events occur at a rate that increases steadily over time
   • Events occur singly, independently and at a constant average rate

4. For X ~ Exp(lambda), which expression gives P(X > x)?

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

5. In the exponential distribution, what does the parameter lambda represent?

   • The median of the distribution
   • The number of events in the interval
   • The rate of events per unit time
   • The mean of the distribution directly

6. Which situation is most naturally modelled by an exponential distribution?

   • The time until the next call arrives at a constant rate
   • The score on a fixed test out of 100
   • The number of heads in 10 coin tosses
   • The heights of students in a class

7. Which statement about an exponential distribution with rate lambda is correct?

   • The pdf is 1 for all x
   • It can take negative values
   • Its cdf is decreasing
   • The total area under the pdf is 1

8. For an exponential distribution with rate lambda = 0.5, what is P(X <= 2), to 4 decimal places?

   • 0.6321
   • 0.1353
   • 0.3679
   • 0.8647

9. For an exponential distribution with rate lambda = 0.5, what is P(X > 4), to 4 decimal places?

   • 0.0183
   • 0.1353
   • 0.6321
   • 0.8647

10. For an exponential distribution with rate lambda = 0.5, what is P(1 <= X <= 3), to 4 decimal places?

    • 0.6321
    • 0.6065
    • 0.2231
    • 0.3834

11. For an exponential distribution with rate lambda = 2 per minute, what is P(X > 1), to 4 decimal places?

    • 0.8647
    • 0.0183
    • 0.6065
    • 0.1353

12. Calls arrive at a rate of 3 per hour. Modelling the waiting time X in hours as exponential, what is P(X <= 1/3), to 4 decimal places?

    • 0.0498
    • 0.9502
    • 0.6321
    • 0.3679

13. For an exponential distribution with rate lambda = 0.5, what is the median?

    • about 1.386
    • about 0.693
    • 4
    • 2

14. For an exponential distribution with rate lambda = 0.5, what is F(4), to 4 decimal places?

    • 0.8647
    • 0.1353
    • 0.9502
    • 0.6321

15. For an exponential distribution with rate lambda = 0.5, the value x satisfies P(X > x) = 0.2. What is x, to 2 decimal places?

    • 6.44
    • 1.61
    • 0.69
    • 3.22

16. For an exponential distribution with rate lambda, P(X > 5 | X > 3) is required. Using the memoryless property, what is this probability when lambda = 0.5, to 4 decimal places?

    • 0.3679
    • 0.2231
    • 0.1353
    • 0.6065

17. A random variable has pdf f(x) = 2e^(-2x) for x >= 0. What is P(X <= 0.5), to 4 decimal places?

    • 0.8647
    • 0.1353
    • 0.6321
    • 0.3679

18. A random variable has pdf f(x) = 3e^(-3x) for x >= 0. What is P(X < 1), to 4 decimal places?

    • 0.6321
    • 0.3679
    • 0.0498
    • 0.9502

19. An exponential random variable has rate lambda = 1.5. What is P(1 <= X <= 2), to 4 decimal places?

    • 0.0498
    • 0.2231
    • 0.1733
    • 0.4231

20. An exponential random variable X has rate lambda. Which integral gives P(a <= X <= b)?

    • Integral of lambda e^(-lambda x) from a to b
    • Integral of e^(-lambda x) from a to b
    • Integral of x lambda e^(-lambda x) from a to b
    • Integral of lambda x from a to b

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