5.04a-b Linear combinations of random variables Quiz: OCR Further Maths, Unit 2

The 20 questions

1. What is the value of E(3X + 4) if E(X) is equal to 5?

   • 27
   • 15
   • 23
   • 19

2. What is the variance of 3X + 4 if the variance of X equals 2?

   • 6
   • 22
   • 18
   • 10

3. If X and Y are independent, what is the formula for the variance of X minus Y?

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

4. If X1 and X2 are independent, identically distributed variables, what is Var(X1 + X2) divided by Var(X)?

   • 0.5
   • 4
   • 1
   • 2

5. For any random variable X, what is the variance of 2X in terms of Var(X)?

   • Var(X)
   • 8 Var(X)
   • 4 Var(X)
   • 2 Var(X)

6. If X follows Po(2) and independent Y follows Po(3), what distribution does X + Y follow?

   • Po(5)
   • Po(1)
   • Po(6)
   • Po(13)

7. If X follows B(10, 0.4) and independent Y follows B(15, 0.4), what distribution does X + Y follow?

   • B(25, 0.4)
   • B(150, 0.4)
   • B(150, 0.16)
   • B(25, 0.8)

8. What is the expected value of the sample mean of n independent observations from a population with mean μ?

   • μ
   • μ / sqrt(n)
   • μ / n
   • n μ

9. What is the variance of the sample mean of n independent observations from a population with variance σ²?

   • σ²
   • n σ²
   • σ / n
   • σ² / n

10. If X follows N(50, 16), what distribution does the sample mean of 4 independent observations follow?

    • N(50, 64)
    • N(50, 16)
    • N(12.5, 4)
    • N(50, 4)

11. If X follows N(10, 4) and Y follows N(5, 9), what is the mean of 2X - Y?

    • 25
    • 15
    • 5
    • 10

12. If independent X follows N(10, 4) and Y follows N(5, 9), what is the variance of 2X - Y?

    • 1
    • 7
    • 31
    • 25

13. What property must two random variables X and Y have for Var(X + Y) to equal Var(X) + Var(Y)?

    • Equal variances
    • Zero mean
    • Independence
    • Identical distributions

14. If the standard deviation of X is 3, what is the standard deviation of -4X + 5?

    • -7
    • -12
    • 17
    • 12

15. If X1, X2, and X3 each have mean 4, what is the expectation of their sum?

    • 16
    • 64
    • 12
    • 4

16. If X1, X2, and X3 are independent variables each with variance 5, what is Var(X1 + X2 + X3)?

    • 45
    • 125
    • 5
    • 15

17. If E(X) = 2 and E(Y) = -3, what is the expected value of 4X - 2Y + 1?

    • 11
    • 3
    • 15
    • 13

18. What term describes the standard deviation of the sample mean estimator?

    • Standard error
    • Pooled deviation
    • Standard bias
    • Sample variance

19. If X and Y are independent Poisson variables, is X - Y also a Poisson variable?

    • Never
    • If means match
    • Always
    • If means differ

20. Any linear combination of independent Normal random variables always follows which distribution?

    • Binomial
    • Poisson
    • Normal
    • Uniform

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