5.03a-d Continuous random variables and probability density functions Quiz: OCR Further Maths, Unit 2

The 20 questions

1. What is the total area under a valid probability density function curve equal to?

   • 1
   • 0.5
   • 0
   • Infinity

2. What condition must a probability density function f(x) satisfy for all x?

   • f(x) <= 0
   • f(x) <= 1
   • f(x) = 1
   • f(x) >= 0

3. Which operation converts a continuous cumulative distribution function F(x) into its probability density function f(x)?

   • Factorisation
   • Substitution
   • Differentiation
   • Integration

4. What is the probability of a continuous random variable taking a single exact value?

   • 0.5
   • Infinity
   • 1
   • 0

5. What value of m satisfies F(m) = 0.5 for a continuous cumulative distribution function?

   • Variance
   • Median
   • Mean
   • Mode

6. How is upper quartile Q3 defined using a continuous cumulative distribution function F(x)?

   • F(Q3) = 3
   • F(Q3) = 0.50
   • F(Q3) = 0.25
   • F(Q3) = 0.75

7. What value of x maximises the probability density function f(x)?

   • Median
   • Range
   • Mean
   • Mode

8. If f(x) = k for x between 0 and 4, what is the value of k?

   • 1
   • 0.5
   • 0.25
   • 4

9. A continuous random variable X has CDF F(x) = x^2 / 9 for 0 <= x <= 3. Find P(X <= 2).

   • 1/3
   • 4/9
   • 2/9
   • 2/3

10. If f(x) = 2x for 0 <= x <= 1, what is the expected value E(X)?

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

11. For a continuous variable X, if E(X) = 3 and E(X^2) = 13, what is Var(X)?

    • 4
    • 10
    • 16
    • 7

12. If Var(X) = 5, what is the variance of 3X + 2?

    • 17
    • 15
    • 45
    • 47

13. A continuous variable X has CDF F(x). What expression gives P(1 <= X <= 4)?

    • F(4) - F(1)
    • F(4) / F(1)
    • F(4) + F(1)
    • F(3)

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

    • 6x
    • 3x^3
    • x^3 / 3
    • x^3

15. If f(x) = 0.5x for 0 <= x <= 2, what is the mode of X?

    • 0
    • 0.5
    • 2
    • 1

16. If f(x) = c x^3 for 0 <= x <= 2, what is the value of c?

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

17. A distribution has CDF F(x) = x^3 for 0 <= x <= 1. What is its median m?

    • 0.75
    • 0.125
    • 0.5
    • (0.5)^(1/3)

18. If f(x) = 2 - 2x for 0 <= x <= 1, what is E(X^2)?

    • 1/4
    • 1/3
    • 1/12
    • 1/6

19. For CDF F(x) = (x - 1)^2 for 1 <= x <= 2, what is f(x)?

    • x - 1
    • 2x + 2
    • 2x - 2
    • 2x

20. How is the lower quartile Q1 defined using a continuous cumulative distribution function F(x)?

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

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