AP Statistics Flashcards: Combining Random Variables
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 11 cards to help you master important concepts.
If E(X) = 10 and E(Y) = 20, calculate the mean of the combined random variable Z = 3X - Y.
The mean is E(Z) = 3*E(X) - 1*E(Y) = 3(10) - 20 = 10.
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If E(X) = 10 and E(Y) = 20, calculate the mean of the combined random variable Z = 3X - Y.
The mean is E(Z) = 3*E(X) - 1*E(Y) = 3(10) - 20 = 10.
The mean and standard deviation of X are 50 and 8, respectively. If Y = 100 + 2X, find the mean and standard deviation of Y.
The new mean is 100 + 2*(50) = 200. The new standard deviation is |2|*8 = 16.
How does a linear transformation (Y = a + bX) affect the shape of a random variable's probability distribution?
A linear transformation does not change the shape of the probability distribution.
Define independent random variables.
Two random variables are independent if knowing the value of one does not change the probability distribution of the other.
What is the formula for the mean of a linear combination of random variables, E(aX + bY)?
The mean of a linear combination is E(aX + bY) = a*mean(X) + b*mean(Y). This formula applies whether the variables are independent or not.
What crucial condition must be met to add or subtract the variances of random variables?
The random variables must be independent to combine their variances.
Independent variables X and Y have variances of 4 and 5, respectively. Calculate the variance of Z = 2X + 3Y.
The variance is Var(Z) = 2^2*Var(X) + 3^2*Var(Y) = 4(4) + 9(5) = 16 + 45 = 61.
For a linear transformation Y = a + bX, how is the standard deviation of Y calculated from the standard deviation of X?
The new standard deviation is |b|*sd(X). Only multiplication affects measures of spread, and spread cannot be negative.
For a linear transformation Y = a + bX, how is the mean of Y calculated from the mean of X?
The new mean is calculated as a + b*mean(X). Both addition and multiplication affect the mean.
What is the formula for the variance of a linear combination of *independent* random variables, Var(aX + bY)?
For independent random variables, the variance is Var(aX + bY) = a^2*Var(X) + b^2*Var(Y).
Why is the variance of the difference of two independent random variables, Var(X - Y), equal to Var(X) + Var(Y)?
Variances add because squaring the coefficient (-1) makes it positive: Var(X - Y) = 1^2*Var(X) + (-1)^2*Var(Y) = Var(X) + Var(Y).