Topic 4.9 Combining Random Variables: apply the rules for the mean and variance of a linear transformation and of sums and differences of random variables, adding variances (not standard deviations) for independent variables.

What this topic is asking

The College Board (Topic 4.9) wants you to apply the rules for the mean and variance of a linear transformation ($aX + b$) and of sums and differences of random variables, in particular that for independent variables you add the variances (never the standard deviations).

Linear transformations

Two things happen separately. Adding a constant $b$ slides every value (and the mean) up by $b$ but leaves the spread unchanged, because shifting everything equally does not change how spread out the values are. Multiplying by $a$ stretches the values, so both the mean and the standard deviation scale by $a$ (the variance by $a^2$). This is why converting units (such as Celsius to Fahrenheit) changes the standard deviation by the multiplier but the added constant does not.

Sums and differences

The mean rule is intuitive: the average of a sum is the sum of the averages, with no independence needed. The variance rule is the one to drill: for independent variables you combine the variances, then square-root to get the standard deviation. Because $3^2 + 4^2 = 25$ gives $\sqrt{25} = 5$, not $3 + 4 = 7$, adding standard deviations directly overstates the spread.

Why variances add, even for a difference

The result that surprises students most is that the variance of a difference is the sum of the variances, the same as for a sum. The reason is that variance measures variability, and combining two independent random quantities makes the result more variable whether you add or subtract them, because each one's fluctuations contribute uncertainty regardless of the sign in front. Subtracting $Y$ does not cancel $Y$'s variability; it still injects it. (The minus sign affects the mean, which is why $\mu_{X-Y} = \mu_X - \mu_Y$, but not the spread.) This matters enormously in inference: the standard deviation of a difference of two independent sample means or proportions (Topics 5.6 and 5.8) is built by adding the two variances and square-rooting, which is exactly this rule. The independence condition is essential, the add-the-variances formula requires $X$ and $Y$ to be independent, so on the exam you should state that the variables are independent before using it.

Putting the rules together

Many problems chain these rules: scale a variable, then add it to another, or sum several independent copies. The reliable approach is to handle the mean and the variance separately, applying the right rule at each step, and only convert to a standard deviation at the very end by square-rooting the final variance. For $n$ independent copies of the same variable $X$, the sum has mean $n\mu_X$ and variance $n\sigma_X^2$ (so standard deviation $\sqrt{n}\,\sigma_X$), which previews why a sample mean's standard deviation shrinks by $\sqrt{n}$ in Unit 5. Keeping the bookkeeping in variances until the end avoids the cardinal error of adding standard deviations partway through. This separation, means add linearly, variances add (for independent variables) and only then take the root, is the master skill of Topic 4.9 and the computational backbone of the sampling-distribution and inference units.

Try this

Q1. Independent $X$ and $Y$ have $\sigma_X = 6$, $\sigma_Y = 8$. Find $\sigma_{X-Y}$. [2 points]

• Cue. Variances add for a difference: $\sigma_{X-Y}^2 = 6^2 + 8^2 = 100$, so $\sigma_{X-Y} = 10$.

Q2. A variable $W$ has sd $4$. Find the sd of $2W + 5$. [1 point]

• Cue. Adding $5$ does not change spread; multiplying by $2$ doubles the sd: $\sigma_{2W+5} = 2(4) = 8$.