Lesson 4B.2.2

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

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson 4B.2.2, Relationship between probability density and distribution functions: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 19: Continuous random variables, written with Revision Ninja.

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The 20 questions

  1. Which expression links the pdf f(x) to the cdf F(x) of a continuous random variable?

    • F(x) equals the derivative of f(x) with respect to x
    • f(x) = dF/dx, the derivative of F with respect to x
    • f(x) = F(x) squared
    • f(x) = 1 minus 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 property must every cdf F(x) of a random variable have?

    • F is always an even function
    • F is non-decreasing
    • F is non-increasing
    • F is always negative

  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. Why is the pdf of a continuous random variable different from its cdf?

    • The cdf is only defined for discrete variables, while the pdf is defined for continuous ones
    • The pdf gives P(X = x) directly, while the cdf cannot be found by integration at all
    • The pdf is the derivative of the cdf and gives density, while the cdf gives cumulative probability
    • The pdf is the integral of the cdf, so the two are reciprocals of each other

  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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