AP Statistics 4.9 Combining Random Variables Study Notes
AP Statistics 4.9 Combining Random Variables Study Notes- New syllabus
AP Statistics 4.9 Combining Random Variables Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- Probability distributions may be used to model variation in populations.
Key Concepts:
- Combining Random Variables
- Linear Transformations of Random Variables
Combining Random Variables
Combining Random Variables
When working with two or more random variables, we often need to determine the combined mean and standard deviation. These rules apply when the random variables are independent.
Mean of Combined Random Variables
The mean of the sum or difference is the sum or difference of the means.
- Formula: \( \mu_{X+Y} = \mu_X + \mu_Y \).
- Formula: \( \mu_{X-Y} = \mu_X – \mu_Y \).
- Interpretation: Expected value behaves linearly.
Variance and Standard Deviation of Combined Random Variables
If \( X \) and \( Y \) are independent, variances add whether combining with a plus or minus.
- Formula: \( \sigma^2_{X+Y} = \sigma^2_X + \sigma^2_Y \).
- Formula: \( \sigma^2_{X-Y} = \sigma^2_X + \sigma^2_Y \).
Standard deviation: \( \sigma_{X \pm Y} = \sqrt{\sigma^2_X + \sigma^2_Y} \).
Important:
You cannot simply add or subtract standard deviations , you must work with variances first.
Example
Suppose the time it takes Student A to finish a test is modeled by random variable \( X \) with mean \( \mu_X = 40 \) minutes. Student B’s test time is modeled by random variable \( Y \) with mean \( \mu_Y = 50 \) minutes.
Find the mean of the combined time \( X + Y \).
▶️ Answer / Explanation
Step 1: Use rule for combined means: \( \mu_{X+Y} = \mu_X + \mu_Y \).
Step 2: \( \mu_{X+Y} = 40 + 50 = 90 \).
Answer: On average, the total time for both students is 90 minutes.
Example
Suppose Student A’s test time has standard deviation \( \sigma_X = 5 \) minutes, and Student B’s test time has standard deviation \( \sigma_Y = 8 \) minutes. Assume \( X \) and \( Y \) are independent.
Find the standard deviation of the total test time \( X + Y \).
▶️ Answer / Explanation
Step 1: Variances add: \( \sigma^2_{X+Y} = \sigma^2_X + \sigma^2_Y \).
Step 2: \( \sigma^2_{X+Y} = 5^2 + 8^2 = 25 + 64 = 89 \).
Step 3: \( \sigma_{X+Y} = \sqrt{89} \approx 9.43 \).
Answer: The standard deviation of the total time is about 9.43 minutes.
Combining Random Variables
Linear Transformations of Random Variables
When we apply a linear transformation to a random variable, such as multiplying by a constant or adding a constant, the mean and standard deviation are affected in predictable ways.
Effects on the Mean
If \( Y = a + bX \), then \( \mu_Y = a + b\mu_X \).
- Adding a constant \( a \) shifts the mean but does not change variability.
- Multiplying by a constant \( b \) stretches or shrinks the mean by factor \( b \).
Effects on the Variance and Standard Deviation
If \( Y = a + bX \), then \( \sigma^2_Y = b^2\sigma^2_X \).
- Variance is affected only by multiplication, not addition.
- Standard deviation: \( \sigma_Y = |b|\sigma_X \).
- Adding a constant does not change spread; multiplying scales spread by \( |b| \).
Example
A random variable \( X \) has mean \( \mu_X = 50 \) and standard deviation \( \sigma_X = 10 \). Define \( Y = X + 5 \).
Find the new mean and standard deviation of \( Y \).
▶️ Answer / Explanation
Step 1: Mean: \( \mu_Y = \mu_X + 5 = 50 + 5 = 55 \).
Step 2: Standard deviation: \( \sigma_Y = \sigma_X = 10 \) (unchanged).
Answer: The mean increases to 55, but the standard deviation remains 10.
Example
A random variable \( X \) has mean \( \mu_X = 50 \) and standard deviation \( \sigma_X = 10 \). Define \( Y = 2X \).
Find the new mean and standard deviation of \( Y \).
▶️ Answer / Explanation
Step 1: Mean: \( \mu_Y = 2 \cdot \mu_X = 2 \cdot 50 = 100 \).
Step 2: Standard deviation: \( \sigma_Y = |2| \cdot 10 = 20 \).
Answer: The mean doubles to 100, and the standard deviation also doubles to 20.
Example
A random variable \( X \) has mean \( \mu_X = 30 \) and standard deviation \( \sigma_X = 4 \). Define \( Y = 3X + 10 \).
Find the new mean and standard deviation of \( Y \).
▶️ Answer / Explanation
Step 1: Mean: \( \mu_Y = 3 \cdot 30 + 10 = 100 \).
Step 2: Standard deviation: \( \sigma_Y = |3| \cdot 4 = 12 \).
Answer: The mean becomes 100, and the standard deviation becomes 12.