SD1-SD3 Probability density functions, medians and quartiles Quiz: AQA Further Maths, Unit 4

The 20 questions

1. For a continuous random variable with probability density function f(x), which two conditions must hold?

   • f(x) is non-negative everywhere and the total integral is 1
   • f(x) is an integer and sums to 1
   • f(x) is at most 1 and the area is 0
   • f(x) is strictly decreasing and integrates to 0

2. For a continuous random variable X, which expression gives P(a <= X <= b)?

   • The integral of x f(x) from a to b
   • f(a) times (b - a)
   • f(b) - f(a)
   • The integral of f(x) from a to b

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

   • Zero
   • The median
   • f(a)
   • 1 divided by the range

4. What does the value of f(x) represent for a continuous random variable?

   • The mean of X
   • The cumulative probability up to x
   • The density of X at x, with probabilities given by areas
   • The probability that X equals x

5. For a continuous random variable, how is the median m defined?

   • The value m with P(X = m) = 0.5
   • The value m with P(X <= m) = 0.5
   • The value m with f(m) = 0.5
   • The value m that maximises f(x)

6. How are the quartiles Q1 and Q3 of a continuous random variable defined?

   • Values with cumulative probability 0.25 and 0.75
   • Values where f(x) equals 0.25 and 0.75
   • Values at one quarter and three quarters of the range
   • Values with probability 0.25 and 0.75 at a point

7. A continuous random variable has pdf f(x) = 2x for 0 <= x <= 1 and zero otherwise. What is P(0.2 <= X <= 0.5)?

   • 0.25
   • 0.04
   • 0.21
   • 0.30

8. For the pdf f(x) = 2x on [0, 1], what is the median?

   • sqrt(0.5), about 0.707
   • 0.25
   • 0.5
   • 0.75

9. For the pdf f(x) = 2x on [0, 1], what is the lower quartile Q1?

   • 0.75
   • sqrt(0.5), about 0.707
   • 0.25
   • 0.5

10. For the pdf f(x) = 2x on [0, 1], what is the upper quartile Q3?

    • 0.707
    • 0.75
    • 0.6
    • 0.866

11. What value of k makes f(x) = kx on [0, 2] (zero elsewhere) a valid pdf?

    • 0.5
    • 2
    • 0.25
    • 1

12. A continuous random variable has pdf f(x) = 3x^2 on [0, 1]. What is P(X <= 0.5)?

    • 0.375
    • 0.5
    • 0.25
    • 0.125

13. A continuous random variable has pdf f(x) = 1/4 on [0, 4] and zero elsewhere. What is its median?

    • 1
    • 2
    • 2.5
    • 4

14. A continuous random variable has pdf f(x) = 2(1 - x) on [0, 1]. What is P(X >= 0.5)?

    • 0.25
    • 0.125
    • 0.75
    • 0.5

15. A continuous random variable has pdf f(x) = 2x on [0, 1]. What is P(X > 0.9)?

    • 0.81
    • 0.1
    • 0.19
    • 0.9

16. A continuous random variable has pdf f(x) = k(4 - x) on [0, 4] and zero elsewhere. What is k?

    • 1/8
    • 1/4
    • 1/2
    • 1/16

17. A continuous random variable has pdf f(x) = k x^2 on [0, 3] and zero elsewhere. What is P(X <= 1)?

    • 1/9
    • 1/3
    • 1/81
    • 1/27

18. For the pdf f(x) = 3x^2 on [0, 1], what is the median?

    • about 0.794
    • 0.5
    • about 0.707
    • 1/3

19. A continuous random variable has pdf f(x) = x on [0, 1] and f(x) = 2 - x on [1, 2], zero elsewhere. What is P(X <= 1)?

    • 0.5
    • 1
    • 0.25
    • 0.75

20. A continuous random variable has pdf f(x) = x on [0, 1] and f(x) = 2 - x on [1, 2], zero elsewhere. Which statement is correct?

    • The pdf is negative on [1, 2]
    • The total area is 2, so f is not a valid pdf
    • The total area is 0.5, so f is not a valid pdf
    • The total area is 1, so f is a valid pdf

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