#AP Statistics: Random Variables - Your Last-Minute Guide 🚀

Hey there, future AP Stats superstar! Let's get you prepped and confident for tomorrow's exam. We're diving into random variables, and I'll keep it concise, clear, and engaging. Let's do this!

#What is a Random Variable?

Think of a random variable as a numerical outcome from a random event. It can be discrete (countable values) or continuous (any value within a range). This section focuses on discrete random variables. Remember, a parameter describes a population, and we use random variables to understand the distribution of these parameters. 🧞‍♂️

Image credit: College Board

#Center of a Discrete Random Variable

#Mean or Expected Value (E(X))

The mean (or expected value) is the average outcome over many trials. It's the long-run average. To find it, multiply each outcome by its probability and sum them up:

$$E(X) = \sum x \cdot P(X=x)$$

Example: If X = number of heads in 2 coin flips:

$$E(X) = (0 \cdot 0.25) + (1 \cdot 0.50) + (2 \cdot 0.25) = 1$$

#Variability of a Discrete Random Variable

#Variance (Var(X))

Variance measures how spread out the values of X are around the mean. It's calculated as the average of the squared differences from the mean:

$$Var(X) = \sum (x - E(X))^2 \cdot P(X=x)$$

#Standard Deviation (SD(X))

The standard deviation is the square root of the variance. It's easier to interpret because it's in the same units as X:

$$SD(X) = \sqrt{Var(X)}$$

#Practice Problem

Let's apply what we've learned with a practice problem!

A random variable, X, represents the number of text messages a person receives in a day. The probability distribution of X is shown in the table below:

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

(b) Calculate the variance of X.

(c) Calculate the standard deviation of X.

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

#Answer

(a) To calculate the mean or expected value of X, you would need to multiply each possible value of X by its probability and then add all of the products. Using the formula E(X) = ∑x * P(X=x), you would get:

$$E(X) = (0 \cdot 0.2) + (1 \cdot 0.3) + (2 \cdot 0.4) + (3 \cdot 0.1) = 1.4$$

So, the mean or expected value of X is 1.4.

(b) To calculate the variance of X, you would need to take the sum of the squares of the differences between each possible value of the variable and the mean, weighted by their probabilities. Using the formula Var(X) = ∑(x - E(X))^2 * P(X=x), you would get:

$$Var(X) = (0 - 1.4)^2 \cdot 0.2 + (1 - 1.4)^2 \cdot 0.3 + (2 - 1.4)^2 \cdot 0.4 + (3 - 1.4)^2 \cdot 0.1 = 1.02$$

So, the variance of X is 1.02.

(c) To calculate the standard deviation of X, you would need to take the square root of the variance. Using the formula SD(X) = √Var(X), you would get:

$$SD(X) = \sqrt{1.02} = 1.00995049$$

So, the standard deviation of X is approximately 1.01.

(d) The mean or expected value of X is 1.4, which tells us that, on average, a person receives 1.4 text messages per day. The variance of X is 1.02, which tells us that the values of X tend to vary somewhat from the mean. The standard deviation of X is 1.01, which tells us that the values of X tend to be relatively closely clustered around the mean. This suggests that the number of text messages a person receives in a day tends to be relatively consistent, with relatively few extreme deviations from the mean.

#Final Exam Focus

Okay, here’s what to focus on for the exam:

• High-Value Topics:

  • Calculating and interpreting the mean (expected value) and standard deviation of discrete random variables.
  • Understanding the difference between variance and standard deviation.
  • Applying these concepts in context (like our text message example).

• Common Question Types:

  • Multiple-choice questions asking for the mean, variance, or standard deviation given a probability distribution.
  • Free-response questions requiring you to calculate and interpret these values in a real-world scenario.
  • Questions that combine concepts from different units, such as sampling distributions and probability.

• Last-Minute Tips:

  • Time Management: Don't spend too long on one question. If you're stuck, move on and come back later.
  • Common Pitfalls: Watch out for calculation errors, especially when squaring and taking square roots. Always double-check your work.
  • Strategies: Always show your work, even if you use a calculator. Include formulas, plugged-in values, and the final answer. Interpret your results in the context of the problem.

  Focus on understanding the formulas and being able to apply them in context. Practice problems are key!

#Practice Questions

Here are some practice questions to solidify your understanding:

Practice Question

Multiple Choice Questions

1. A discrete random variable X has the following probability distribution:

X	1	2	3	4
P(X)	0.2	0.3	0.4	0.1

What is the expected value of X?

(A) 2.0 (B) 2.3 (C) 2.5 (D) 3.0 (E) 3.5

2. The variance of a discrete random variable Y is 2.25. What is the standard deviation of Y?

(A) 0.75 (B) 1.125 (C) 1.5 (D) 2.25 (E) 5.06

3. A game involves rolling a six-sided die. If you roll a 1 or 2, you win $5. If you roll a 3, 4, or 5, you lose$2. If you roll a 6, you win $10. What is the expected value of this game?$

(A)1.50 (B) $2.00 (C)$2.50 (D) $3.00 (E)$3.50

Free Response Question

A company sells a product with a warranty. The probability distribution for the number of warranty claims per year is given below:

Claims (X)	0	1	2	3	4
P(X)	0.40	0.30	0.15	0.10	0.05

(a) Calculate the mean (expected value) of the number of warranty claims per year. (b) Calculate the variance of the number of warranty claims per year. (c) Calculate the standard deviation of the number of warranty claims per year. (d) Interpret the mean and standard deviation in the context of the problem.

Scoring Breakdown:

(a) Mean (Expected Value):

• 1 point for correctly multiplying each outcome by its probability.
• 1 point for summing the products correctly.
• 1 point for the correct answer: E(X) = 0(0.40) + 1(0.30) + 2(0.15) + 3(0.10) + 4(0.05) = 1.1

(b) Variance:

• 1 point for correctly calculating the squared differences from the mean.
• 1 point for multiplying the squared differences by their probabilities.
• 1 point for summing the products correctly.
• 1 point for the correct answer: Var(X) = (0-1.1)^2(0.40) + (1-1.1)^2(0.30) + (2-1.1)^2(0.15) + (3-1.1)^2(0.10) + (4-1.1)^2(0.05) = 1.49

(c) Standard Deviation:

• 1 point for correctly taking the square root of the variance.
• 1 point for the correct answer: SD(X) = √1.49 ≈ 1.22

(d) Interpretation:

• 1 point for correctly interpreting the mean in context: On average, the company expects 1.1 warranty claims per year.
• 1 point for correctly interpreting the standard deviation in context: The number of warranty claims typically varies by about 1.22 claims from the mean each year.

You've got this! Go ace that exam!