SD6-SD8 Cumulative distribution functions and the rectangular distribution Quiz: AQA Further Maths, Unit 4

The 20 questions

1. For a continuous random variable X, what does the cumulative distribution function F(x) give?

   • The mean of X up to x
   • P(X = x)
   • P(X <= x)
   • The density of X at x

2. What is the relationship between the pdf f(x) and the cdf F(x) of a continuous random variable?

   • F(x) = d f(x)/dx
   • f(x) = 1 - F(x)
   • f(x) = F(x) squared
   • f(x) = dF/dx

3. Which integral expresses F(x) in terms of f(t)?

   • Integral of f(x) dt from 0 to 1
   • Integral of f(t) from minus infinity to x
   • Integral of t f(t) from minus infinity to x
   • Integral of f(t) from x to infinity

4. Which pair of properties must any cumulative distribution function F(x) satisfy?

   • F(x) is strictly decreasing from 1 to 0
   • F(x) takes negative values only
   • F(x) tends to 1 at minus infinity and 0 at infinity
   • F(x) is non-decreasing with limits 0 and 1

5. What is the pdf of the rectangular distribution on [a, b]?

   • 1/(a + b) for a <= x <= b
   • (b - a) for a <= x <= b
   • 1/(b - a) for a <= x <= b
   • x/(b - a) for a <= x <= b

6. For any two random variables X and Y, which is always true?

   • E(X + Y) = E(X) + E(Y)
   • E(X + Y) = E(X) - E(Y)
   • E(X + Y) = E(X + Y) squared
   • E(X + Y) = E(X) E(Y)

7. For two random variables X and Y, what is needed for Var(X + Y) = Var(X) + Var(Y)?

   • X and Y must be independent
   • X and Y must have the same mean
   • X and Y must both be uniform
   • X and Y must take non-negative values

8. A continuous random variable has F(x) = x^2 for 0 <= x <= 1. What is P(X <= 0.5)?

   • 0.75
   • 0.25
   • 0.125
   • 0.5

9. A continuous random variable has F(x) = x^2 for 0 <= x <= 1. What is its pdf on [0, 1]?

   • 2
   • x^2
   • x
   • 2x

10. A continuous random variable X has cdf F(x) = (x - 1)/3 for 1 <= x <= 4. What is P(2 <= X <= 3)?

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

11. A random variable X is uniform on [2, 8]. What is E(X)?

    • 3
    • 4
    • 6
    • 5

12. A random variable X is uniform on [2, 8]. What is Var(X)?

    • 3
    • 9
    • 6
    • 12

13. A random variable X is uniform on [0, 10]. What is P(X > 7)?

    • 0.5
    • 0.7
    • 0.3
    • 0.2

14. Independent random variables X and Y have Var(X) = 3 and Var(Y) = 5. What is Var(X + Y)?

    • 4
    • 8
    • 15
    • 2

15. Random variables X and Y have E(X) = 2 and E(Y) = -1. What is E(X + Y)?

    • 3
    • -3
    • 1
    • 2

16. A continuous random variable has cdf F(x) = 3x^2 - 2x^3 on [0, 1]. What is its pdf?

    • 6x
    • 6x^2
    • 6x(1 - x)
    • 3x^2

17. A rectangular distribution on [a, b] has variance 4/3. What is b - a?

    • sqrt(4/3)
    • 4
    • 2
    • 16

18. Independent X is uniform on [0, 1] and Y is uniform on [0, 2]. What is Var(X + Y)?

    • 5/12
    • 1/4
    • 1/3
    • 2/3

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

    • 0.5
    • 0.125
    • 0.375
    • 0.875

20. A continuous random variable has pdf f(x) = 1/4 on [0, 4]. What is F(2)?

    • 2
    • 0.5
    • 0.75
    • 0.25

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