AP Statistics · Topic 4.9

Combining Random Variables Practice

Part of Probability, Random Variables, and Probability Distributions.(VAR-5.C)

Practice questions

18

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Sample questions

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Sample 1difficulty 2/5
$X has mean 20, variance 9. Let W = 2X.$
$What is the variance of W?$
- A
  $18$
- B
  $81$
- C
  $36$
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- D
  $9$
Why
$Var(aX) = a^2 Var(X) = 4 * 9 = 36.$
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Sample 2difficulty 2/5
$X has mean 50 and SD 8. Let Y = X + 10.$
$What is mean and SD of Y?$
- A
  $mean 50, SD 8$
- B
  $mean 60, SD 18$
- C
  $mean 60, SD 80$
- D
  $mean 60, SD 8$
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Why
$Adding constant shifts mean by 10 but does not change SD.$
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Sample 3difficulty 3/5
$X and Y are independent with means 5 and 8, variances 4 and 9.$
$What is the SD of X + Y?$
- A
  $5$
- B
  $sqrt(5)$
- C
  $13$
- D
  $sqrt(13)$
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Why
$Var(X+Y) = Var(X) + Var(Y) = 4 + 9 = 13. SD = sqrt(13).$
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Sample 4difficulty 3/5
$X has mean 10 and SD 2. Let Y = 3X + 5.$
$What is the mean and SD of Y?$
- A
  $mean 30, SD 6$
- B
  $mean 35, SD 11$
- C
  $mean 35, SD 2$
- D
  $mean 35, SD 6$
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Why
$E(Y) = 3(10) + 5 = 35. SD(Y) = 3 * SD(X) = 6 (constants don't affect SD).$
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Sample 5difficulty 3/5
$For RVs A, B with means 30 and 25, mean of A - B is$
- A
  $5$
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- B
  $Cannot tell$
- C
  $55$
- D
  $-5$
Why
$E(A - B) = 30 - 25 = 5. Holds even without independence.$
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