Lesson 4B.2.1

4B.2.1 Probability density 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.2.1, Probability density 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 two conditions must the probability density function f(x) of a continuous random variable satisfy?

    • f(x) equals P(X = x) for each x, and the sum of f(x) over all x equals 1
    • f(x) is always increasing, and f(x) equals 1 at the mean of X
    • f(x) is never negative, and the integral of f(x) over all x equals 1
    • f(x) is never greater than 1, and the integral of f(x) over all x equals 0

  2. For a continuous random variable X, what is P(X = c) for any single value c?

    • 1 divided by the number of values that X can take
    • F(c), the cumulative distribution function evaluated at c
    • f(c), the value of the probability density function at c
    • 0, because the probability is an area under the curve over a zero-width interval

  3. Which expression gives P(a < X <= b) for a continuous random variable X with pdf f(x)?

    • f(b) minus f(a)
    • The integral of x f(x) with respect to x from a to b
    • f(a) multiplied by (b minus a)
    • The integral of f(x) with respect to x from a to b

  4. A continuous random variable has pdf f(x) = 1/4 for 2 <= x <= 6 and 0 otherwise. What is P(X > 5)?

    • 0.25
    • 0.75
    • 0.5
    • 0.4

  5. The pdf of a continuous random variable can take values greater than 1 at some points. Which statement explains why this is allowed?

    • Only the total area under the pdf must equal 1, so individual heights can exceed 1
    • The pdf must be symmetric about the mean, so heights above 1 are always allowed
    • The pdf gives probabilities directly, so each height must lie between 0 and 1
    • The pdf must always be at most 1 for every x, so this is impossible

  6. Which of the following is a valid pdf on the interval [0, 1]?

    • f(x) = 2 - x on [0, 1], since its height is positive throughout
    • f(x) = 3x^2 on [0, 2], since it is never negative and its height is at most 12
    • f(x) = 3x^2 on [0, 1], since its integral over [0, 1] is 1
    • f(x) = x^2 on [0, 1], since it is never negative

  7. For a pdf f, what does the integral from minus infinity to x0 of f(t) dt represent?

    • The probability that X equals x0 exactly
    • P(X <= x0), the cumulative distribution function at x0
    • The value of the pdf at x0
    • The mean of X for all values up to x0

  8. For a continuous random variable with cdf F, which expression gives P(X > a)?

    • the value F(a) itself
    • 1 minus the density value f(a)
    • 1 minus F(a)
    • the density value f(a)

  9. A continuous random variable has pdf f(x) = k x^2 for 0 <= x <= 2 and 0 otherwise. What is the value of k?

    • 1/2
    • 3/4
    • 3/8
    • 1/8

  10. A continuous random variable has pdf f(x) = (3/8) x^2 for 0 <= x <= 2 and 0 otherwise. What is P(X < 1)?

    • 1/8
    • 3/8
    • 1/3
    • 1/2

  11. A continuous random variable has pdf f(x) = 2x for 0 <= x <= 1 and 0 otherwise. What is P(X > 0.5)?

    • 0.375
    • 0.25
    • 0.5
    • 0.75

  12. A continuous random variable has pdf f(x) = (3/4)(1 - x^2) for -1 <= x <= 1 and 0 otherwise. What is P(0 < X < 1)?

    • 3/4
    • 1/2
    • 1/4
    • 3/8

  13. A continuous random variable has pdf f(x) = x/8 for 0 <= x <= 4 and 0 otherwise. What is P(X > 3)?

    • 3/8
    • 1/4
    • 7/16
    • 1/2

  14. A continuous random variable has pdf f(x) = c e^(-x) for x >= 0 and 0 otherwise. What is the value of c?

    • e
    • 1
    • 1/e
    • 2

  15. A continuous random variable has pdf f(x) = k(x + 1) for 0 <= x <= 2 and 0 otherwise. What is the value of k?

    • 1/4
    • 1/6
    • 1/3
    • 1/2

  16. A continuous random variable has pdf f(x) = (3/32) x(4 - x) for 0 <= x <= 4 and 0 otherwise. What is P(X > 2)?

    • 1/2
    • 3/8
    • 5/8
    • 1/4

  17. A continuous random variable has pdf f(x) = 4x^3 for 0 <= x <= 1 and 0 otherwise. What is P(X > 1/2)?

    • 7/8
    • 1/2
    • 1/16
    • 15/16

  18. A continuous random variable has pdf f(x) = kx for 0 <= x <= 2 and f(x) = k(4 - x) for 2 < x <= 4, zero otherwise. What is k?

    • 1/2
    • 1/4
    • 1/8
    • 1/3

  19. A continuous random variable has pdf f(x) = (3/8) x^2 for 0 <= x <= 2 and 0 otherwise. What is P(X > 1 given X > 1/2)?

    • 8/9
    • 7/8
    • 1/2
    • 63/64

  20. A continuous random variable has pdf f(x) = kx for 0 <= x <= 1 and f(x) = k(2 - x) for 1 < x <= 2, zero otherwise. What is k?

    • 1
    • 2
    • 1/4
    • 1/2

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