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 model assumes events occur independently and singly at a constant average rate?

   • Binomial distribution
   • Uniform distribution
   • Normal distribution
   • Exponential distribution

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 an exponential distribution with parameter lambda, what does lambda represent?

   • Variance of time
   • Probability of success
   • Rate of events
   • Mean waiting time

6. Which variable is most naturally modelled using an exponential distribution?

   • Time between calls
   • Height of adults
   • Score on dice
   • Number of calls

7. What is the total area under the probability density function of an exponential distribution?

   • 1 / lambda
   • lambda
   • Infinity
   • 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. For an exponential distribution with rate lambda, what is the probability density function?

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

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