4B.2.1 Probability density functions Quiz: Pearson Edexcel Further Maths, Unit 19

The 20 questions

1. What must the integral of a pdf f(x) over all x equal?

   • Infinity
   • 0.5
   • 1
   • 0

2. What is P(X = c) for a continuous random variable X at a specific point c?

   • 0.5
   • f(c)
   • 1
   • 0

3. Which operation on pdf f(x) from a to b gives P(a < X <= b)?

   • Differentiation
   • Indefinite integration
   • Evaluation at b
   • Definite integration

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. What is the maximum value a probability density function f(x) can take?

   • No upper bound
   • 0
   • 0.5
   • 1

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

   • f(x) = 4x^2
   • f(x) = x^2
   • f(x) = 3x^2
   • f(x) = 2x^2

7. What does the integral of a pdf f(t) from minus infinity to x represent?

   • Variance Var(X)
   • Cumulative distribution F(x)
   • Expected value E(X)
   • Probability density f(x)

8. For a continuous random variable with cdf F(x), what is P(X > a)?

   • F(a)
   • 1 + F(a)
   • 1 - F(a)
   • 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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