AP Stats AP Stats Random Variables and Expected Value 1 — Questions and Answers
Question 1: The expected value (mean) of a discrete random variable X is calculated as:
- Σ x / n
- Σ x · P(X = x) (Correct answer)
- Σ P(X = x)
- max(x) − min(x)
Correct answer: Σ x · P(X = x)
The expected value E(X) = Σ x · P(X = x), the probability-weighted sum of all possible values.
Question 2: A fair coin is flipped 3 times. Let X = number of heads. What is E(X)?
- 1
- 1.5 (Correct answer)
- 2
- 0.5
Correct answer: 1.5
X ~ Binomial(n=3, p=0.5), so E(X) = np = 3 × 0.5 = 1.5.
Question 3: If X is a random variable and a and b are constants, then E(aX + b) equals:
- aE(X) + b (Correct answer)
- aE(X)
- E(X) + b
- a²E(X) + b
Correct answer: aE(X) + b
By linearity of expectation, E(aX + b) = aE(X) + b.
Question 4: Which of the following describes a continuous random variable?
- Number of students absent each day
- Score on a 40-question multiple-choice test
- Exact time in seconds for a runner to finish a race (Correct answer)
- Number of cars in a parking lot
Correct answer: Exact time in seconds for a runner to finish a race
A continuous random variable can take any value in an interval; exact time is continuous, while counts are discrete.
Question 5: The variance of a discrete random variable X is defined as:
- E(X²)
- E(X) − [E(X)]²
- E[(X − μ)²] (Correct answer)
- E(X) × P(X)
Correct answer: E[(X − μ)²]
Var(X) = E[(X − μ)²] = the expected squared deviation of X from its mean.
Question 6: If X and Y are independent random variables, then Var(X + Y) equals:
- Var(X) − Var(Y)
- Var(X) × Var(Y)
- Var(X) + Var(Y) (Correct answer)
- [Var(X) + Var(Y)] / 2
Correct answer: Var(X) + Var(Y)
For independent random variables, variances add: Var(X + Y) = Var(X) + Var(Y).
AP Stats AP Stats Random Variables and Expected Value 1
The expected value (mean) of a discrete random variable X is calculated as: