AP Statistics (AP Stats): Combining Independent Random Variables (Mean and SD of Sums/Differences)

AP Statistics – Combining Independent Random Variables (Mean and SD of Sums/Differences)

AP Statistics Study Guide: Combining Independent Random Variables (Mean and SD of Sums/Differences)

What This Is

This topic explains how to find the mean (expected value) and standard deviation (SD) of the sum or difference of two (or more) independent random variables. It’s essential for the AP exam because it allows you to analyze real-world scenarios like: - A factory produces two types of widgets; what’s the total expected weight and variability if you combine one of each? - A student takes two exams; what’s the expected difference in their scores, and how much could that difference vary? - A company’s revenue comes from two independent products; what’s the total expected revenue and its uncertainty?

Mastering this helps you solve FRQs on probability, sampling distributions, and confidence intervals where combining variables is required.

Key Terms & Formulas

• Independent Random Variables: Two variables are independent if knowing the value of one gives no information about the other (e.g., flipping a coin and rolling a die).
• Mean (Expected Value) of a Sum: E(X + Y) = E(X) + E(Y)
• The expected value of the sum is the sum of the expected values.
• Mean (Expected Value) of a Difference: E(X – Y) = E(X) – E(Y)
• The expected value of the difference is the difference of the expected values.
• Variance of a Sum (Independent Variables): Var(X + Y) = Var(X) + Var(Y)
• Variances add only if the variables are independent.
• Variance of a Difference (Independent Variables): Var(X – Y) = Var(X) + Var(Y)
• Variance of a difference is the same as variance of a sum (because variance is always positive).
• Standard Deviation of a Sum/Difference: SD(X ± Y) = ?(Var(X) + Var(Y))
• Take the square root of the variance to get the SD.
• Linear Combinations of Random Variables: For constants a and b, E(aX + bY) = a·E(X) + b·E(Y) and Var(aX + bY) = a²·Var(X) + b²·Var(Y).
• 10% Condition (for Independence in Sampling): If sampling without replacement, check that the sample size n-10% of the population to assume independence.
• Normal Approximation for Sums/Differences: If X and Y are both normally distributed, then X + Y and X – Y are also normally distributed.
• Central Limit Theorem (CLT) for Sums: Even if X and Y are not normal, the sum (or difference) of many independent variables will be approximately normal if the sample size is large enough.

Step-by-Step / Process Flow

Follow these steps for a typical FRQ on combining independent random variables:

1. Identify the Variables and Their Distributions
2. Define X and Y (e.g., X = score on Exam 1, Y = score on Exam 2).
3. Note their means (, ) and standard deviations (, ).

4. Check if they are independent (if not, you can’t combine variances!).

5. Determine What You’re Combining (Sum or Difference?)

6. Are you asked for X + Y (e.g., total score) or X – Y (e.g., difference in scores)?

7. Calculate the Mean of the Combined Variable

8. For sum: E(X + Y) = +

9. For difference: E(X – Y) = –

10. Calculate the Variance of the Combined Variable

11. Only if independent! Var(X ± Y) = ² + ²

12. If not independent, you cannot combine variances (this is a common trap!).

13. Find the Standard Deviation

14. SD(X ± Y) = ?(² + ²)

15. Check for Normality (If Needed)

16. If X and Y are normal, X ± Y is normal.
17. If not, check if the CLT applies (large sample size).

18. If normal, use normalcdf( on TI-84 for probabilities.

19. Answer the Question in Context

20. Example: “The expected total score is 170 points, with a standard deviation of 12 points.”

Common Mistakes

• Mistake: Adding standard deviations instead of variances.
• Correction: SD(X + Y)- + . You must add variances first, then take the square root.

• Why? Variance is additive for independent variables, but SD is not.

• Mistake: Forgetting to check independence before combining variances.

• Correction: If X and Y are not independent, you cannot use Var(X + Y) = Var(X) + Var(Y).

• Why? Dependence introduces covariance, which affects variance.

• Mistake: Assuming X – Y has a different variance than X + Y.

• Correction: Var(X – Y) = Var(X) + Var(Y) (same as sum).

• Why? Variance is always positive, so subtracting doesn’t change it.

• Mistake: Ignoring the 10% condition when sampling without replacement.

• Correction: If sampling from a finite population, check n-0.10N to assume independence.

• Why? Without replacement, selections are not independent unless the sample is small relative to the population.

• Mistake: Using the wrong distribution for probabilities.

• Correction: Only use the normal distribution if X and Y are normal or the sample size is large (CLT).
• Why? Without normality or large n, the sum/difference may not be normal.

AP Exam Insights

• FRQs often ask for:
• The mean and SD of a sum/difference (e.g., total revenue, difference in test scores).
• Probabilities involving sums/differences (e.g., “What’s the probability the total weight exceeds 500 lbs?”).

• Linear combinations (e.g., 2X + 3Y).

• Tricky Distinctions:

• Sum vs. Difference Variance: Var(X – Y) = Var(X) + Var(Y), not Var(X) – Var(Y).
• Independence is crucial: If the problem doesn’t state independence, you cannot combine variances.

• Normality matters: If X and Y are not normal, the sum/difference may not be normal unless n is large.

• Calculator Tips:

• Use normalcdf(lower, upper, ?, ?) for probabilities of sums/differences.

• For linear combinations, first compute the new mean and SD, then use normalcdf.

• Common FRQ Setup:

• A problem gives you two independent random variables (e.g., X = weight of a bag of chips, Y = weight of a soda can) and asks for the mean/SD of their sum or difference.
• Then, it asks for a probability (e.g., “What’s the probability the total weight is less than 16 oz?”).

Quick Check Questions

Question 1 (Multiple Choice)

A factory produces two types of batteries. Battery A has a mean lifetime of 100 hours with a standard deviation of 10 hours. Battery B has a mean lifetime of 120 hours with a standard deviation of 15 hours. The lifetimes are independent. What is the standard deviation of the difference in lifetimes (A – B)?

(A) 5 hours (B) ?(10² + 15²) hours (C) 10 + 15 hours (D) ?(10² – 15²) hours (E) 15 – 10 hours

Correct Answer: (B) ?(10² + 15²) hours Explanation: Var(A – B) = Var(A) + Var(B) = 10² + 15², so SD(A – B) = ?(10² + 15²).

Question 2 (FRQ Part)

A college student’s commute consists of two parts: X = time spent on the bus (mean = 20 minutes, SD = 5 minutes) and Y = time spent walking (mean = 10 minutes, SD = 2 minutes). The times are independent.

a) What is the mean and standard deviation of the total commute time (X + Y)? b) What is the probability that the total commute time exceeds 35 minutes? Assume X + Y is normally distributed.

Correct Answers: a) Mean = 30 minutes, SD = ?(5² + 2²) = ?29-5.39 minutes - E(X + Y) = 20 + 10 = 30 - Var(X + Y) = 5² + 2² = 29-SD = ?29

b) P(X + Y > 35)-0.180 (using normalcdf(35, 1E99, 30, ?29) on TI-84) - Z = (35 – 30) / ?29-0.928-P(Z > 0.928)-0.1766

Last-Minute Cram Sheet

1. Mean of Sum/Difference: E(X ± Y) = E(X) ± E(Y)
2. Variance of Sum/Difference (Independent): Var(X ± Y) = Var(X) + Var(Y)
3. SD of Sum/Difference: SD(X ± Y) = ?(Var(X) + Var(Y))
4. Linear Combinations: E(aX + bY) = a·E(X) + b·E(Y), Var(aX + bY) = a²·Var(X) + b²·Var(Y)
5. Normality Check: If X and Y are normal, X ± Y is normal. If not, check CLT.
6. 10% Condition: For sampling without replacement, n-0.10N to assume independence.
7. Calculator Command: normalcdf(lower, upper, ?, ?) for probabilities.
8. Never add SDs directly! Always add variances first.
9. Independence is required to combine variances.
10. Variance of difference = variance of sum (always add variances).