Combining Random Variables - AP Statistics
Card 1 of 30
Find the mean of $X + Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$.
Find the mean of $X + Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$.
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$\text{E}(X + Y) = 8$. Apply linearity: $5 + 3 = 8$.
$\text{E}(X + Y) = 8$. Apply linearity: $5 + 3 = 8$.
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State the formula for the mean of the difference of two random variables $X$ and $Y$.
State the formula for the mean of the difference of two random variables $X$ and $Y$.
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$\text{E}(X - Y) = \text{E}(X) - \text{E}(Y)$. Expected value is linear, so differences distribute across expectations.
$\text{E}(X - Y) = \text{E}(X) - \text{E}(Y)$. Expected value is linear, so differences distribute across expectations.
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What is the mean of the difference $X - Y$ given $\text{E}(X) = 7$ and $\text{E}(Y) = 2$?
What is the mean of the difference $X - Y$ given $\text{E}(X) = 7$ and $\text{E}(Y) = 2$?
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$\text{E}(X - Y) = 5$. Apply difference formula: $7 - 2 = 5$.
$\text{E}(X - Y) = 5$. Apply difference formula: $7 - 2 = 5$.
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What is the mean of $4X - Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$?
What is the mean of $4X - Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$?
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$\text{E}(4X - Y) = 17$. Apply linear combination: $4(5) - 1(3) = 20 - 3 = 17$.
$\text{E}(4X - Y) = 17$. Apply linear combination: $4(5) - 1(3) = 20 - 3 = 17$.
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What is the variance of $4X - Y$ given $\text{Var}(X) = 2$ and $\text{Var}(Y) = 1$, assuming independence?
What is the variance of $4X - Y$ given $\text{Var}(X) = 2$ and $\text{Var}(Y) = 1$, assuming independence?
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$\text{Var}(4X - Y) = 33$. Calculate: $4^2(2) + 1^2(1) = 32 + 1 = 33$.
$\text{Var}(4X - Y) = 33$. Calculate: $4^2(2) + 1^2(1) = 32 + 1 = 33$.
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What is the variance of $3X$ given $\text{Var}(X) = 5$?
What is the variance of $3X$ given $\text{Var}(X) = 5$?
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$\text{Var}(3X) = 45$. Apply variance scaling: $3^2(5) = 9(5) = 45$.
$\text{Var}(3X) = 45$. Apply variance scaling: $3^2(5) = 9(5) = 45$.
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What is the formula for the standard deviation of $aX$?
What is the formula for the standard deviation of $aX$?
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$\text{SD}(aX) = |a|\text{SD}(X)$. Standard deviation scales by absolute value of the constant.
$\text{SD}(aX) = |a|\text{SD}(X)$. Standard deviation scales by absolute value of the constant.
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What is the mean of the sum $X + Y + Z$ given $\text{E}(X) = 3$, $\text{E}(Y) = 4$, and $\text{E}(Z) = 5$?
What is the mean of the sum $X + Y + Z$ given $\text{E}(X) = 3$, $\text{E}(Y) = 4$, and $\text{E}(Z) = 5$?
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$\text{E}(X + Y + Z) = 12$. Sum all expected values: $3 + 4 + 5 = 12$.
$\text{E}(X + Y + Z) = 12$. Sum all expected values: $3 + 4 + 5 = 12$.
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What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, and $\text{Var}(Z) = 3$, assuming independence?
What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, and $\text{Var}(Z) = 3$, assuming independence?
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$\text{Var}(X + Y + Z) = 6$. Sum all variances: $1 + 2 + 3 = 6$.
$\text{Var}(X + Y + Z) = 6$. Sum all variances: $1 + 2 + 3 = 6$.
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What is the formula for the mean of a random variable $bY$?
What is the formula for the mean of a random variable $bY$?
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$\text{E}(bY) = b\text{E}(Y)$. Expected value scales linearly with multiplication by constants.
$\text{E}(bY) = b\text{E}(Y)$. Expected value scales linearly with multiplication by constants.
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What is the variance of $2Y$ given $\text{Var}(Y) = 4$?
What is the variance of $2Y$ given $\text{Var}(Y) = 4$?
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$\text{Var}(2Y) = 16$. Apply variance scaling: $2^2(4) = 4(4) = 16$.
$\text{Var}(2Y) = 16$. Apply variance scaling: $2^2(4) = 4(4) = 16$.
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What is the standard deviation of $2Y$ given $\text{SD}(Y) = 3$?
What is the standard deviation of $2Y$ given $\text{SD}(Y) = 3$?
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$\text{SD}(2Y) = 6$. Apply standard deviation scaling: $2(3) = 6$.
$\text{SD}(2Y) = 6$. Apply standard deviation scaling: $2(3) = 6$.
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What is the mean of the sum $X + Y + Z$ given $\text{E}(X) = 3$, $\text{E}(Y) = 4$, and $\text{E}(Z) = 5$?
What is the mean of the sum $X + Y + Z$ given $\text{E}(X) = 3$, $\text{E}(Y) = 4$, and $\text{E}(Z) = 5$?
Tap to reveal answer
$\text{E}(X + Y + Z) = 12$. Sum all expected values: $3 + 4 + 5 = 12$.
$\text{E}(X + Y + Z) = 12$. Sum all expected values: $3 + 4 + 5 = 12$.
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What is the variance of $2Y$ given $\text{Var}(Y) = 4$?
What is the variance of $2Y$ given $\text{Var}(Y) = 4$?
Tap to reveal answer
$\text{Var}(2Y) = 16$. Apply variance scaling: $2^2(4) = 4(4) = 16$.
$\text{Var}(2Y) = 16$. Apply variance scaling: $2^2(4) = 4(4) = 16$.
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What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, and $\text{Var}(Z) = 3$, assuming independence?
What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, and $\text{Var}(Z) = 3$, assuming independence?
Tap to reveal answer
$\text{Var}(X + Y + Z) = 6$. Sum all variances: $1 + 2 + 3 = 6$.
$\text{Var}(X + Y + Z) = 6$. Sum all variances: $1 + 2 + 3 = 6$.
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What is the standard deviation of $2Y$ given $\text{SD}(Y) = 3$?
What is the standard deviation of $2Y$ given $\text{SD}(Y) = 3$?
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$\text{SD}(2Y) = 6$. Apply standard deviation scaling: $2(3) = 6$.
$\text{SD}(2Y) = 6$. Apply standard deviation scaling: $2(3) = 6$.
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What is the variance of $4X - Y$ given $\text{Var}(X) = 2$ and $\text{Var}(Y) = 1$, assuming independence?
What is the variance of $4X - Y$ given $\text{Var}(X) = 2$ and $\text{Var}(Y) = 1$, assuming independence?
Tap to reveal answer
$\text{Var}(4X - Y) = 33$. Calculate: $4^2(2) + 1^2(1) = 32 + 1 = 33$.
$\text{Var}(4X - Y) = 33$. Calculate: $4^2(2) + 1^2(1) = 32 + 1 = 33$.
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What is the formula for the mean of the sum of two random variables $X$ and $Y$?
What is the formula for the mean of the sum of two random variables $X$ and $Y$?
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$\text{E}(X + Y) = \text{E}(X) + \text{E}(Y)$. Expected value is linear, so sums distribute across expectations.
$\text{E}(X + Y) = \text{E}(X) + \text{E}(Y)$. Expected value is linear, so sums distribute across expectations.
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What is the variance of $X + Y$ given $\text{Var}(X) = 1$ and $\text{Var}(Y) = 4$, assuming independence?
What is the variance of $X + Y$ given $\text{Var}(X) = 1$ and $\text{Var}(Y) = 4$, assuming independence?
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$\text{Var}(X + Y) = 5$. For independent variables: $1 + 4 = 5$.
$\text{Var}(X + Y) = 5$. For independent variables: $1 + 4 = 5$.
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What is the mean of the difference $X - Y$ given $\text{E}(X) = 7$ and $\text{E}(Y) = 2$?
What is the mean of the difference $X - Y$ given $\text{E}(X) = 7$ and $\text{E}(Y) = 2$?
Tap to reveal answer
$\text{E}(X - Y) = 5$. Apply difference formula: $7 - 2 = 5$.
$\text{E}(X - Y) = 5$. Apply difference formula: $7 - 2 = 5$.
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What is the mean of $4X - Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$?
What is the mean of $4X - Y$ given $\text{E}(X) = 5$ and $\text{E}(Y) = 3$?
Tap to reveal answer
$\text{E}(4X - Y) = 17$. Apply linear combination: $4(5) - 1(3) = 20 - 3 = 17$.
$\text{E}(4X - Y) = 17$. Apply linear combination: $4(5) - 1(3) = 20 - 3 = 17$.
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What is the formula for the variance of $X + Y + Z$ assuming independence?
What is the formula for the variance of $X + Y + Z$ assuming independence?
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$\text{Var}(X + Y + Z) = \text{Var}(X) + \text{Var}(Y) + \text{Var}(Z)$. Variances add for independent variables regardless of quantity.
$\text{Var}(X + Y + Z) = \text{Var}(X) + \text{Var}(Y) + \text{Var}(Z)$. Variances add for independent variables regardless of quantity.
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What is the mean of $2X + 3Y + 4Z$ given $\text{E}(X) = 1$, $\text{E}(Y) = 2$, $\text{E}(Z) = 3$?
What is the mean of $2X + 3Y + 4Z$ given $\text{E}(X) = 1$, $\text{E}(Y) = 2$, $\text{E}(Z) = 3$?
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$\text{E}(2X + 3Y + 4Z) = 20$. Apply linear combination: $2(1) + 3(2) + 4(3) = 2 + 6 + 12 = 20$.
$\text{E}(2X + 3Y + 4Z) = 20$. Apply linear combination: $2(1) + 3(2) + 4(3) = 2 + 6 + 12 = 20$.
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What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, $\text{Var}(Z) = 3$, assuming independence?
What is the variance of $X + Y + Z$ given $\text{Var}(X) = 1$, $\text{Var}(Y) = 2$, $\text{Var}(Z) = 3$, assuming independence?
Tap to reveal answer
$\text{Var}(X + Y + Z) = 6$. Sum all variances: $1 + 2 + 3 = 6$.
$\text{Var}(X + Y + Z) = 6$. Sum all variances: $1 + 2 + 3 = 6$.
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What is the mean of $5X - 2Y$ given $\text{E}(X) = 2$ and $\text{E}(Y) = 3$?
What is the mean of $5X - 2Y$ given $\text{E}(X) = 2$ and $\text{E}(Y) = 3$?
Tap to reveal answer
$\text{E}(5X - 2Y) = 4$. Apply linear combination: $5(2) - 2(3) = 10 - 6 = 4$.
$\text{E}(5X - 2Y) = 4$. Apply linear combination: $5(2) - 2(3) = 10 - 6 = 4$.
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What is the mean of $2Y$ given $\text{E}(Y) = 6$?
What is the mean of $2Y$ given $\text{E}(Y) = 6$?
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$\text{E}(2Y) = 12$. Apply scaling: $2(6) = 12$.
$\text{E}(2Y) = 12$. Apply scaling: $2(6) = 12$.
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What is the formula for the variance of a random variable $aX$?
What is the formula for the variance of a random variable $aX$?
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$\text{Var}(aX) = a^2\text{Var}(X)$. Variance scales by the square of the constant multiplier.
$\text{Var}(aX) = a^2\text{Var}(X)$. Variance scales by the square of the constant multiplier.
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State the formula for the mean of the difference of two random variables $X$ and $Y$.
State the formula for the mean of the difference of two random variables $X$ and $Y$.
Tap to reveal answer
$\text{E}(X - Y) = \text{E}(X) - \text{E}(Y)$. Expected value is linear, so differences distribute across expectations.
$\text{E}(X - Y) = \text{E}(X) - \text{E}(Y)$. Expected value is linear, so differences distribute across expectations.
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What is the formula for the variance of a linear combination $aX + bY$ when $X$ and $Y$ are independent?
What is the formula for the variance of a linear combination $aX + bY$ when $X$ and $Y$ are independent?
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$\text{Var}(aX + bY) = a^2\text{Var}(X) + b^2\text{Var}(Y)$. Coefficients are squared when calculating variance of linear combinations.
$\text{Var}(aX + bY) = a^2\text{Var}(X) + b^2\text{Var}(Y)$. Coefficients are squared when calculating variance of linear combinations.
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What is the formula for the mean of a linear combination $aX + bY$?
What is the formula for the mean of a linear combination $aX + bY$?
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$\text{E}(aX + bY) = a\text{E}(X) + b\text{E}(Y)$. Linearity of expectation applies to any linear combination.
$\text{E}(aX + bY) = a\text{E}(X) + b\text{E}(Y)$. Linearity of expectation applies to any linear combination.
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