4B.2.2 Relationship between probability density and distribution functions Quiz: Pearson Edexcel Further Maths, Unit 19

The 20 questions

1. How is the pdf f(x) obtained from the cdf F(x) of a continuous random variable?

   • Integrate F(x)
   • Differentiate F(x)
   • Square F(x)
   • Invert F(x)

2. For a cdf F of a continuous random variable, what are the limits of F(x) as x tends to minus infinity and as x tends to infinity?

   • 0 and infinity
   • 0 and 1
   • minus 1 and 1
   • minus infinity and 0

3. Which mathematical property must every cumulative distribution function F(x) satisfy?

   • Bounded by 1
   • Non-decreasing
   • Strictly decreasing
   • Symmetric

4. How is P(a < X <= b) written in terms of the cdf F?

   • F(b) minus F(a)
   • F(b) multiplied by F(a)
   • F(a) minus F(b)
   • F(b) plus F(a)

5. The pdf of X is 0 outside [0, 1]. For 0 <= x <= 1 the cdf is F(x) = x^2. What is f(x) on this interval?

   • 2x
   • x^2
   • 2
   • x

6. The cdf of Y is F(y) = 0 for y < 0, F(y) = y/5 for 0 <= y <= 5, and 1 for y > 5. What is the pdf for 0 <= y <= 5?

   • 5
   • 1/5
   • 1/25
   • y/5

7. What does the derivative of the cumulative distribution function F(x) represent?

   • Cumulative probability
   • Variance
   • Probability density
   • Expected value

8. A cdf is F(x) = x^2/16 for 0 <= x <= 4. What is P(X > 2)?

   • 3/4
   • 3/8
   • 1/4
   • 1/2

9. A continuous random variable has F(x) = x^3 on [0, 1]. What is P(0.5 < X <= 0.8)?

   • 0.625
   • 0.512
   • 0.387
   • 0.300

10. A continuous random variable has F(x) = 1 - e^(-2x) for x >= 0. What is P(X > 1), to 3 decimal places?

    • 0.135
    • 0.607
    • 0.865
    • 0.271

11. A continuous random variable has F(x) = (x - 1)/3 on [1, 4]. What is P(2 < X <= 3)?

    • 1/6
    • 1/2
    • 2/3
    • 1/3

12. A continuous random variable has cdf F(x) = 1 - (1 + x) e^(-x) for x >= 0. What is the value of the pdf f(1)?

    • e, about 2.718
    • 2/e, about 0.736
    • 1/(2e), about 0.184
    • 1/e, about 0.368

13. A continuous random variable has F(x) = x^2/9 on [0, 3]. Find the median of X.

    • sqrt(4.5), about 2.12
    • 2.25
    • 1.5
    • 1.06

14. A cdf has the form F(x) = a x^2 + b x on [0, 1], with F(0) = 0 and F(1) = 1. Given that the pdf satisfies f(0) = 1/2, which pair (a, b) is correct?

    • a = 1, b = 0
    • a = 1/2, b = 1/2
    • a = 1/4, b = 3/4
    • a = 0, b = 1

15. A continuous random variable has cdf F(x) = (1 - cos x)/2 on [0, pi]. What is P(X < pi/3)?

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

16. A continuous random variable has cdf F(x) = x^2/16 on [0, 4]. Find the value of x such that P(X > x) = 0.9.

    • 2.00, from x^2 = 4
    • 0.40, from x^2 = 0.16
    • 1.26, from x^2 = 1.6
    • 3.16, from x^2 = 10

17. A continuous random variable has cdf F(x) = x^2/4 on [0, 2] and F(x) = 0 or 1 outside. What is P(X > 1 given X < 1.5)?

    • 5/16
    • 4/9
    • 5/9
    • 1/2

18. A continuous random variable has F(x) = x^2 on [0, 1]. What is P(X < 0.8 given X > 0.5)?

    • 0.52
    • 0.39
    • 0.64
    • 0.75

19. A continuous random variable has cdf F(x) = (1 - cos x)/2 on [0, pi]. What is the pdf f(x) on this interval?

    • sin(x)
    • (1 + cos x)/2
    • sin(x)/2
    • cos(x)/2

20. The cdf of X is F(x) = x^2/16 on [0, 4]. Find the pdf f(x) on [0, 4].

    • x/16
    • x^2/8
    • 1/8
    • x/8

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