Transforming and Combining Random Variables 📊

Hey there, future AP Stats pro! Let's dive into transforming and combining random variables. This is a super useful skill that'll pop up all over the exam. Think of it as your secret weapon for simplifying calculations and making sense of data. Let's get started!

Linear Transformations of a Random Variable

Adding or Subtracting a Constant ➕➖

When you add or subtract a constant from a random variable, you're basically shifting the entire distribution along the number line. This affects the center and location but not the spread or the shape.

• Mean: The mean shifts by the same constant. If Y = X + c, then E(Y) = E(X) + c.
• Standard Deviation: The standard deviation stays the same. SD(Y) = SD(X).

Multiplying or Dividing by a Constant ✖️➗

Multiplying or dividing by a constant changes the center, location, and spread of the distribution but not the shape.

• Mean: The mean is multiplied or divided by the same constant. If Y = c * X, then E(Y) = c * E(X).
• Standard Deviation: The standard deviation is also multiplied or divided by the same constant. SD(Y) = |c| * SD(X).

Y = a + BX

Combining Random Variables

Sometimes, you'll need to combine two or more random variables. Here's how to handle it:

Expected Value of the Sum/Difference of Two Random Variables

💡 Summary

• Sum: If S = X + Y, then E(S) = E(X) + E(Y). The mean of the sum is the sum of the means.
• Difference: If D = X - Y, then E(D) = E(X) - E(Y). The mean of the difference is the difference of the means. Order matters!.

Standard Deviation of the Sum/Difference of Two Random Variables

💡 Summary

• Sum: If S = X + Y, then Var(S) = Var(X) + Var(Y), and SD(S) = √(Var(X) + Var(Y)).
• Difference: If D = X - Y, then Var(D) = Var(X) + Var(Y), and SD(D) = √(Var(X) + Var(Y)).

Practice Problem

Let's put this into practice! 😉

Two random variables, X and Y, represent the number of hours a student spends studying for a math test and the number of hours a student spends studying for a science test, respectively. The probability distributions of X and Y are shown in the tables below:

A new random variable, Z, represents the total number of hours a student spends studying for both tests.

(a) Calculate the mean or expected value of Z.

(b) Calculate the standard deviation of Z.

(c) Interpret the results in the context of the problem.

Practice Question

Practice Questions

Multiple Choice Questions

1. Let X be a random variable with mean μ and standard deviation σ. If Y = 2X + 3, what is the mean and standard deviation of Y? (A) Mean: μ + 3, Standard Deviation: σ + 3 (B) Mean: 2μ + 3, Standard Deviation: 2σ (C) Mean: 2μ + 3, Standard Deviation: 2σ + 3 (D) Mean: 2μ + 3, Standard Deviation: 4σ (E) Mean: 2μ + 3, Standard Deviation: 4σ + 3

2. Random variables X and Y are independent. If X has a mean of 10 and a standard deviation of 2, and Y has a mean of 15 and a standard deviation of 3, what is the standard deviation of X + Y? (A) 1 (B) 5 (C) √13 (D) √5 (E) √3

3. If the random variable X has a mean of 20 and a standard deviation of 5, and the random variable Y has a mean of 10 and a standard deviation of 3, what is the mean of 2X - Y? (A) 10 (B) 20 (C) 30 (D) 40 (E) 50

Free Response Question

A company manufactures two types of components, A and B, for electronic devices. The time it takes to produce each component is a random variable. The production time for component A has a mean of 5 minutes and a standard deviation of 1 minute. The production time for component B has a mean of 8 minutes and a standard deviation of 2 minutes. Assume the production times for components A and B are independent.

(a) What is the mean and standard deviation of the total time to produce one of each component (A and B)?

(b) A device requires two units of component A and one unit of component B. What is the mean and standard deviation of the total time to produce the components needed for one device?

(c) If the company produces 100 devices, what is the mean and standard deviation of the total time to produce all the components for these 100 devices?

Scoring Guide

(a) Mean: E(A + B) = E(A) + E(B) = 5 + 8 = 13 minutes. Standard Deviation: SD(A + B) = √((1)^2 + (2)^2) = √5 ≈ 2.236 minutes. 1 point for correct mean, 1 point for correct standard deviation.

(b) Mean: E(2A + B) = 2E(A) + E(B) = 2(5) + 8 = 18 minutes. Standard Deviation: SD(2A + B) = √((2*1)^2 + (2)^2) = √8 ≈ 2.828 minutes. 1 point for correct mean, 1 point for correct standard deviation.

(c) Mean: For 100 devices, the total mean is 100 * 18 = 1800 minutes. Standard Deviation: For 100 devices, the total standard deviation is √100 * 2.828 = 10 * 2.828 = 28.28 minutes. 1 point for correct mean, 1 point for correct standard deviation.

Final Exam Focus

Okay, you're almost there! Here's what to focus on for the exam:

• Transformations: Understand how adding/subtracting and multiplying/dividing constants affect the mean and standard deviation. Remember, only multiplication/division changes the spread.
• Combining Variables: Know how to find the mean and standard deviation of sums and differences of independent random variables. Variances add, standard deviations don't!
• Independence: Always check if random variables are independent before combining them. This is a key assumption for using the variance rules.

High-Value Topic: Questions involving linear transformations and combinations of random variables are very common on the AP exam. Make sure you are comfortable with these concepts and formulas.

You've got this! Take a deep breath, review these notes, and go ace that exam! 🎉