AP Statistics (AP Stats): Discrete Random Variables (Mean, Variance, Standard Deviation)

AP Statistics – Discrete Random Variables (Mean, Variance, Standard Deviation)

AP Statistics: Discrete Random Variables (Mean, Variance, Standard Deviation) – Exam-Ready Study Guide

What This Is

Discrete random variables assign numerical outcomes to events with countable possibilities (e.g., number of heads in 10 coin flips, number of defective items in a batch). On the AP exam, you’ll calculate and interpret the mean (expected value), variance, and standard deviation of discrete random variables, often using probability distributions. These concepts are foundational for understanding sampling distributions, hypothesis tests, and confidence intervals. Real-world example: A factory produces light bulbs with a 2% defect rate. If you inspect 50 bulbs, how many defects do you expect to find, and how much variability is there in that count?

Key Terms & Formulas

• Discrete random variable (X): A variable that takes on a countable number of distinct values (e.g., number of successes in n trials).
• Probability distribution: A table, graph, or formula listing all possible values of X and their probabilities (P(X)). Must satisfy:
• 0-P(X)-1 for all X.
• ?P(X) = 1 (sum of all probabilities = 1).
• Mean (Expected Value) of X ( or E(X)): = ?[x · P(x)]
• x = value of the random variable.
• P(x) = probability of x.
• Interpretation: The long-run average value of X after many repetitions.
• Variance of X (²): ² = ?[(x – )² · P(x)] or ² = ?[x² · P(x)] – ² (shortcut formula).
• Measures the spread of X around its mean.
• Standard Deviation of X (): = ?²
• Interpretation: The typical distance of X from its mean (in the same units as X).
• Linear Transformations of X (Y = aX + b):
• Mean: = a + b
• Variance: ² = a²² (adding b doesn’t affect spread).
• Standard Deviation: = |a|
• Combining Independent Random Variables (X and Y):
• Mean: = + ; = –
• Variance: ² = ² + ²; ² = ² + ² (variances add even for subtraction).
• Standard Deviation: = ?(² + ²)
• Binomial Random Variable (X ~ B(n, p)):
• Conditions (BINS):
  • Binary outcomes (success/failure).
  • Independent trials.
  • Number of trials (n) is fixed.
  • Same probability of success (p) for each trial.
• Mean: = np
• Variance: ² = np(1 – p)
• Standard Deviation: = ?[np(1 – p)]
• Calculator: binompdf(n, p, k) for P(X = k); binomcdf(n, p, k) for P(X-k).
• Geometric Random Variable (X ~ G(p)):
• Conditions: Same as binomial, but trials continue until the first success.
• Mean: = 1/p
• Variance: ² = (1 – p)/p²
• Standard Deviation: = ?[(1 – p)/p²]
• Calculator: geometpdf(p, k) for P(X = k); geometcdf(p, k) for P(X-k).

Step-by-Step / Process Flow

How to solve an AP FRQ on discrete random variables:
1. Identify the random variable (X): - Define X in context (e.g., "Let X = number of defective bulbs in a sample of 50"). - Determine if X is binomial, geometric, or another discrete distribution.
2. Write the probability distribution: - For binomial/geometric: State n and p (or just p for geometric). - For general discrete: List all possible values of X and their probabilities.
3. Calculate the mean (expected value): - Use = ?[x · P(x)] or the shortcut formula for binomial/geometric. - Interpret: "In the long run, we expect [] [units] on average."
4. Calculate variance and standard deviation: - Use ² = ?[(x – )² · P(x)] or the shortcut ² = ?[x² · P(x)] – ². - For binomial/geometric, use the formulas above. - Interpret SD: "The number of [X] typically varies by about [] from the mean of []."
5. Answer the question in context: - If asked for P(X = k), use binompdf/geometpdf or the probability distribution. - If asked for P(X-k), use binomcdf/geometcdf. - For transformations (e.g., Y = 2X + 3), apply the rules for mean/variance/SD.

Common Mistakes

• Mistake: Forgetting to check the 10% condition for binomial distributions when sampling without replacement.

• Correction: If the sample size n is >10% of the population, the trials aren’t independent. Use the binomial distribution only if n-0.1N (where N = population size).

• Mistake: Using the wrong formula for variance (e.g., mixing up binomial variance with general variance).

• Correction: For binomial, ² = np(1 – p). For general discrete, use ?[(x – )² · P(x)].

• Mistake: Misinterpreting the mean as a guaranteed outcome.

• Correction: The mean is a long-run average. For example, if = 3.2 defects, you might get 2 or 4 defects in a single sample, but the average over many samples is 3.2.

• Mistake: Adding standard deviations when combining random variables.

• Correction: Variances add for independent variables, not standard deviations. For X and Y, = ?(² + ²).

• Mistake: Using binompdf for P(X-k) instead of binomcdf.

• Correction: binompdf(n, p, k) gives P(X = k); binomcdf(n, p, k) gives P(X-k).

AP Exam Insights

• Tricky Distinction: Binomial vs. geometric distributions.
• Binomial: Fixed number of trials (n); count successes.
• Geometric: Trials until first success; count trials needed.
• Common FRQ Setup:
• Given a probability distribution table, calculate mean/variance/SD.
• Given a binomial scenario, find P(X = k) or P(X-k) and interpret the mean/SD.
• Combining two independent random variables (e.g., X = profit from Product A, Y = profit from Product B; find mean/SD of total profit).
• Calculator Pitfalls:
• Mixing up binompdf and binomcdf (or geometpdf and geometcdf).
• Forgetting to square a in the variance transformation rule (² = a²²).
• AP Grader Pet Peeve: Not interpreting answers in context. Always write, "We expect an average of [] [units] in the long run" or "The number of [X] typically varies by about [] from the mean."

Quick Check Questions

1. Multiple Choice: A fair six-sided die is rolled. Let X = the number rolled. What is the standard deviation of X?
2. (A) 1.71
3. (B) 1.87
4. (C) 2.00
5. (D) 3.50

6. Answer: (A) 1.71 Explanation: Calculate = 3.5, then = ?[?(x – 3.5)² · (1/6)]-1.71.

7. FRQ Part: A factory produces light bulbs with a 5% defect rate. In a random sample of 100 bulbs:

8. (a) What is the expected number of defective bulbs?
9. (b) What is the standard deviation of the number of defective bulbs?

10. Answer: (a) = np = 100(0.05) = 5 defective bulbs. (b) = ?[np(1 – p)] = ?[100(0.05)(0.95)]-2.18 defective bulbs.

11. Multiple Choice: Let X and Y be independent random variables with = 4, = 1, = 6, and = 2. What is the standard deviation of X + Y?

12. (A) 1.41
13. (B) 2.24
14. (C) 3.00
15. (D) 5.00
16. Answer: (B) 2.24 Explanation: = ?(² + ²) = ?(1 + 4)-2.24.

Last-Minute Cram Sheet

1. Mean (Expected Value): = ?[x · P(x)] or np (binomial).
2. Variance: ² = ?[(x – )² · P(x)] or np(1 – p) (binomial).
3. Standard Deviation: = ?².
4. Linear Transformations: = a + b; ² = a²²; = |a|.
5. Combining Variables: = + ; ² = ² + ² (if independent).
6. Binomial Conditions (BINS): Binary, Independent, fixed n, Same p.
7. Binomial Mean/SD: = np; = ?[np(1 – p)].
8. Geometric Mean/SD: = 1/p; = ?[(1 – p)/p²].
9. Calculator Commands:
10. binompdf(n, p, k) for P(X = k).
11. binomcdf(n, p, k) for P(X-k).
12. geometpdf(p, k) for P(X = k).
13. geometcdf(p, k) for P(X-k).
14. Traps:
    • Variances add; standard deviations don’t.
    • Always check the 10% condition for binomial distributions.
    • Interpret mean/SD in context (e.g., "expected number," "typical variation").