AP Statistics Flashcards: Combining Random Variables

What this deck covers

This deck focuses on Combining Random Variables, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: What is the formula for the variance of a random variable $aX$?

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

Flashcard 3: What is the formula for the mean of the sum of two random variables $X$ and $Y$?

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

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

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

Flashcard 5: What is the standard deviation of $2Y$ given $\text{SD}(Y) = 3$?

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

Flashcard 6: What is the formula for the standard deviation of $aX$?

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

Flashcard 7: What is the formula for the variance of a random variable $aX$?

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

Flashcard 8: What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, and $\text{Var}(Z) = 3$, assuming independence?

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

Flashcard 9: What is the formula for the variance of the sum of two independent random variables $X$ and $Y$?

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

Flashcard 10: What is the mean of $2Y$ given $\text{E}(Y) = 6$?

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

Flashcard 11: What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, $\text{Var}(Z) = 3$, assuming independence?

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

Flashcard 12: What is the formula for the variance of the sum of two independent random variables $X$ and $Y$?

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

Flashcard 13: What is the formula for the mean of the sum of two random variables $X$ and $Y$?

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

Flashcard 14: What is the formula for the mean of a random variable $aX$?

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

Flashcard 15: What is the variance of $X + Y$ given $\text{Var}(X) = 1$ and $\text{Var}(Y) = 4$, assuming independence?

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

Flashcard 16: What is the formula for the mean of $X + Y + Z$?

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

Flashcard 17: What is the variance of $2Y$ given $\text{Var}(Y) = 4$?

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

Flashcard 18: What is the variance of $X - Y$ given $\text{Var}(X) = 3$ and $\text{Var}(Y) = 2$, assuming independence?

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

Flashcard 19: What is the formula for the variance of a linear combination $aX + bY$ when $X$ and $Y$ are independent?

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

Flashcard 20: What is the mean of the sum $X + Y + Z$ given $\text{E}(X) = 3$, $\text{E}(Y) = 4$, and $\text{E}(Z) = 5$?

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

Flashcard 21: Find the mean of $3X - 2Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$.

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

Flashcard 22: Find the variance of $X + Y$ given $\text{Var}(X) = 4$ and $\text{Var}(Y) = 9$, assuming $X$ and $Y$ are independent.

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

Flashcard 23: What is the mean of $4X - Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$?

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

Flashcard 24: What is the formula for the mean of a linear combination $aX + bY$?

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

Flashcard 25: What is the mean of the sum $X + Y + Z$ given $\text{E}(X) = 3$, $\text{E}(Y) = 4$, and $\text{E}(Z) = 5$?

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

Flashcard 26: Identify the formula for the variance of the difference of two independent random variables $X$ and $Y$.

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

Flashcard 27: What is the formula for the variance of $X + Y + Z$ assuming independence?

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

Flashcard 28: What is the formula for the mean of a linear combination $aX + bY$?

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

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

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

Flashcard 30: What is the formula for the variance of $X + Y + Z$ assuming independence?

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

Flashcard 31: What is the mean of $2X + 3Y + 4Z$ given $\text{E}(X) = 1$, $\text{E}(Y) = 2$, $\text{E}(Z) = 3$?

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

Flashcard 32: What is the formula for the variance of a random variable $bY$?

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

Flashcard 33: What is the mean of $5X - 2Y$ given $\text{E}(X) = 2$ and $\text{E}(Y) = 3$?

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

Flashcard 34: What is the mean of the difference $X - Y$ given $\text{E}(X) = 7$ and $\text{E}(Y) = 2$?

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

Flashcard 35: What is the mean of $5X - 2Y$ given $\text{E}(X) = 2$ and $\text{E}(Y) = 3$?

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

Flashcard 36: What is the variance of $4X - Y$ given $\text{Var}(X) = 2$ and $\text{Var}(Y) = 1$, assuming independence?

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

Flashcard 37: What is the mean of $4X - Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$?

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

Flashcard 38: What is the formula for the mean of a random variable $bY$?

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

Flashcard 39: State the formula for the mean of the difference of two random variables $X$ and $Y$.

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

Flashcard 40: What is the formula for the standard deviation of $aX$?

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

Flashcard 41: What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, $\text{Var}(Z) = 3$, assuming independence?

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

Flashcard 42: What is the mean of the difference $X - Y$ given $\text{E}(X) = 7$ and $\text{E}(Y) = 2$?

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

Flashcard 43: What is the mean of $2X + 3Y + 4Z$ given $\text{E}(X) = 1$, $\text{E}(Y) = 2$, $\text{E}(Z) = 3$?

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

Flashcard 44: What is the variance of $X - Y$ given $\text{Var}(X) = 3$ and $\text{Var}(Y) = 2$, assuming independence?

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

Flashcard 45: What is the variance of $X + Y$ given $\text{Var}(X) = 1$ and $\text{Var}(Y) = 4$, assuming independence?

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

Flashcard 46: What is the variance of $3X$ given $\text{Var}(X) = 5$?

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

Flashcard 47: What is the variance of $4X - Y$ given $\text{Var}(X) = 2$ and $\text{Var}(Y) = 1$, assuming independence?

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

Flashcard 48: What is the mean of $2Y$ given $\text{E}(Y) = 6$?

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

Flashcard 49: What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, and $\text{Var}(Z) = 3$, assuming independence?

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

Flashcard 50: Find the mean of $3X - 2Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$.

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

Flashcard 51: What is the formula for the variance of a random variable $bY$?

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

Flashcard 52: Find the mean of $X + Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$.

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

Flashcard 53: State the formula for the mean of the difference of two random variables $X$ and $Y$.

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

Flashcard 54: What is the standard deviation of $2Y$ given $\text{SD}(Y) = 3$?

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

Flashcard 55: What is the mean of $2X + 3Y$ given $\text{E}(X) = 4$ and $\text{E}(Y) = 6$?

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

Flashcard 56: What is the mean of $3X$ given $\text{E}(X) = 4$?

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