5.03e Cumulative distribution functions Quiz: OCR Further Maths, Unit 2

The 20 questions

1. What mathematical operation converts a cumulative distribution function F(x) into a probability density function f(x)?

   • Factorisation
   • Differentiation
   • Substitution
   • Integration

2. What is the value of the cumulative distribution function F(x) at the upper boundary of support?

   • 0.5
   • 0
   • Infinity
   • 1

3. For a continuous random variable X, how is P(a < X <= b) calculated using its CDF F(x)?

   • F(b) - F(a)
   • F(a) - F(b)
   • F(b) / F(a)
   • F(a) + F(b)

4. What equation must be solved to find the lower quartile Q1 using the CDF F(x)?

   • F(Q1) = 0.25
   • F(Q1) = 0.75
   • F(Q1) = 0.5
   • F(Q1) = 0.1

5. What equation must be solved to find the median m of a continuous random variable using F(x)?

   • F(m) = 1
   • F(m) = 0.5
   • F(m) = 0.75
   • F(m) = 0.25

6. If F(x) = x^2 for 0 <= x <= 1, what is the probability density function f(x)?

   • 2
   • 2x
   • x
   • x^3 / 3

7. If f(x) = 3x^2 for 0 <= x <= 1, what is the cumulative distribution function F(x)?

   • x^2
   • x^3
   • 6x
   • x^4 / 4

8. If F(x) = x / 8 for 0 <= x <= 8, what is the median value of X?

   • 4
   • 2
   • 6
   • 8

9. Which property must all continuous cumulative distribution functions F(x) satisfy across their domain?

   • Non-decreasing
   • Periodic
   • Symmetric
   • Strictly decreasing

10. If Y = X^3 for a positive random variable X, how is F_Y(y) expressed in terms of F_X?

    • F_X(y^3)
    • F_X(y^(1/3))
    • (F_X(y))^3
    • 3 F_X(y)

11. For independent random variables X and Y, what is Var(X - Y) equal to?

    • 2Var(X) - 2Var(Y)
    • Var(X)Var(Y)
    • Var(X) + Var(Y)
    • Var(X) - Var(Y)

12. If E(X) = 4 and E(Y) = 3, what is E(2X - 3Y + 5)?

    • -1
    • 14
    • 4
    • 9

13. If Var(X) = 5, what is the value of Var(3X + 4)?

    • 15
    • 49
    • 45
    • 19

14. If Var(X) = 4 and Var(Y) = 9 for independent X and Y, find Var(2X - Y).

    • 23
    • 25
    • 1
    • 7

15. If F(x) = x^4 for 0 <= x <= 1, what equation gives the 90th percentile P90?

    • (P90)^4 = 0.1
    • 4(P90)^3 = 0.9
    • (P90)^4 = 0.9
    • (P90)^4 = 90

16. What is the value of the cumulative distribution function F(x) for x below the minimum support value?

    • 0.5
    • 0
    • 1
    • -1

17. If X has CDF F(x) = x^2 for 0 <= x <= 1, what is P(0.2 < X <= 0.6)?

    • 0.16
    • 0.36
    • 0.32
    • 0.4

18. If E(X) = 10, what is E(12 - 4X)?

    • 52
    • -40
    • 28
    • -28

19. If Y = 2X + 1 and X has CDF F_X, what is F_Y(y)?

    • F_X((y - 1)/2)
    • F_X((y + 1)/2)
    • F_X(2y + 1)
    • 2 F_X(y) + 1

20. Which calculus operation applied to f(x) yields the cumulative distribution function F(x)?

    • Exponentiation
    • Differentiation
    • Integration
    • Factorisation

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