Combining Random Variables - AP Statistics

1. What is the variance of $2Y$ given $\text{Var}(Y) = 4$?

   $\text{Var}(2Y) = 16$. Apply variance scaling: $2^2(4) = 4(4) = 16$.

2. What is the formula for the mean of the sum of two random variables $X$ and $Y$?

   $\text{E}(X + Y) = \text{E}(X) + \text{E}(Y)$. Expected value is linear, so sums distribute across expectations.

3. What is the mean of $2X + 3Y$ given $\text{E}(X) = 4$ and $\text{E}(Y) = 6$?

   $\text{E}(2X + 3Y) = 26$. Apply linear combination: $2(4) + 3(6) = 8 + 18 = 26$.

4. What is the formula for the variance of a random variable $aX$?

   $\text{Var}(aX) = a^2\text{Var}(X)$. Variance scales by the square of the constant multiplier.

5. What is the formula for the variance of the sum of two independent random variables $X$ and $Y$?

   $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$. For independent variables, variances add when combining random variables.

6. What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, $\text{Var}(Z) = 3$, assuming independence?

   $\text{Var}(X + Y + Z) = 6$. Sum all variances: $1 + 2 + 3 = 6$.

7. What is the formula for the mean of a random variable $aX$?

   $\text{E}(aX) = a\text{E}(X)$. Expected value scales linearly with multiplication by constants.

8. What is the variance of $X + Y$ given $\text{Var}(X) = 1$ and $\text{Var}(Y) = 4$, assuming independence?

   $\text{Var}(X + Y) = 5$. For independent variables: $1 + 4 = 5$.

9. What is the formula for the mean of $X + Y + Z$?

   $\text{E}(X + Y + Z) = \text{E}(X) + \text{E}(Y) + \text{E}(Z)$. Expected value is linear for any number of variables.

10. What is the formula for the variance of a linear combination $aX + bY$ when $X$ and $Y$ are independent?

    $\text{Var}(aX + bY) = a^2\text{Var}(X) + b^2\text{Var}(Y)$. Coefficients are squared when calculating variance of linear combinations.

11. Find the variance of $X + Y$ given $\text{Var}(X) = 4$ and $\text{Var}(Y) = 9$, assuming $X$ and $Y$ are independent.

    $\text{Var}(X + Y) = 13$. For independent variables: $4 + 9 = 13$.

12. What is the formula for the mean of a linear combination $aX + bY$?

    $\text{E}(aX + bY) = a\text{E}(X) + b\text{E}(Y)$. Linearity of expectation applies to any linear combination.

13. What is the mean of the sum $X + Y + Z$ given $\text{E}(X) = 3$, $\text{E}(Y) = 4$, and $\text{E}(Z) = 5$?

    $\text{E}(X + Y + Z) = 12$. Sum all expected values: $3 + 4 + 5 = 12$.

14. Identify the formula for the variance of the difference of two independent random variables $X$ and $Y$.

    $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y)$. Variance of difference equals sum of variances for independent variables.

15. Find the mean of $X + Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$.

    $\text{E}(X + Y) = 8$. Apply linearity: $5 + 3 = 8$.

16. What is the formula for the variance of $X + Y + Z$ assuming independence?

    $\text{Var}(X + Y + Z) = \text{Var}(X) + \text{Var}(Y) + \text{Var}(Z)$. Variances add for independent variables regardless of quantity.

17. What is the formula for the variance of a random variable $bY$?

    $\text{Var}(bY) = b^2\text{Var}(Y)$. Variance scales by the square of the constant multiplier.

18. What is the mean of $5X - 2Y$ given $\text{E}(X) = 2$ and $\text{E}(Y) = 3$?

    $\text{E}(5X - 2Y) = 4$. Apply linear combination: $5(2) - 2(3) = 10 - 6 = 4$.

19. What is the variance of $4X - Y$ given $\text{Var}(X) = 2$ and $\text{Var}(Y) = 1$, assuming independence?

    $\text{Var}(4X - Y) = 33$. Calculate: $4^2(2) + 1^2(1) = 32 + 1 = 33$.

20. What is the mean of $4X - Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$?

    $\text{E}(4X - Y) = 17$. Apply linear combination: $4(5) - 1(3) = 20 - 3 = 17$.

21. What is the formula for the mean of a random variable $bY$?

    $\text{E}(bY) = b\text{E}(Y)$. Expected value scales linearly with multiplication by constants.

22. What is the formula for the standard deviation of $aX$?

    $\text{SD}(aX) = |a|\text{SD}(X)$. Standard deviation scales by absolute value of the constant.

23. What is the mean of the difference $X - Y$ given $\text{E}(X) = 7$ and $\text{E}(Y) = 2$?

    $\text{E}(X - Y) = 5$. Apply difference formula: $7 - 2 = 5$.

24. What is the mean of $2X + 3Y + 4Z$ given $\text{E}(X) = 1$, $\text{E}(Y) = 2$, $\text{E}(Z) = 3$?

    $\text{E}(2X + 3Y + 4Z) = 20$. Apply linear combination: $2(1) + 3(2) + 4(3) = 2 + 6 + 12 = 20$.

25. What is the variance of $X - Y$ given $\text{Var}(X) = 3$ and $\text{Var}(Y) = 2$, assuming independence?

    $\text{Var}(X - Y) = 5$. For independent variables: $3 + 2 = 5$.

26. What is the variance of $3X$ given $\text{Var}(X) = 5$?

    $\text{Var}(3X) = 45$. Apply variance scaling: $3^2(5) = 9(5) = 45$.

27. What is the mean of $2Y$ given $\text{E}(Y) = 6$?

    $\text{E}(2Y) = 12$. Apply scaling: $2(6) = 12$.

28. What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, and $\text{Var}(Z) = 3$, assuming independence?

    $\text{Var}(X + Y + Z) = 6$. Sum all variances: $1 + 2 + 3 = 6$.

29. Find the mean of $3X - 2Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$.

    $\text{E}(3X - 2Y) = 9$. Apply linear combination formula: $3(5) - 2(3) = 15 - 6 = 9$.

30. State the formula for the mean of the difference of two random variables $X$ and $Y$.

    $\text{E}(X - Y) = \text{E}(X) - \text{E}(Y)$. Expected value is linear, so differences distribute across expectations.

31. What is the standard deviation of $2Y$ given $\text{SD}(Y) = 3$?

    $\text{SD}(2Y) = 6$. Apply standard deviation scaling: $2(3) = 6$.

32. What is the mean of $3X$ given $\text{E}(X) = 4$?

    $\text{E}(3X) = 12$. Apply scaling: $3(4) = 12$.

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