AP Stats AP Stats Random Variables and Expected Value

Question 1

The expected value (mean) of a discrete random variable X is calculated as:

Accepted Answer

Σ x · P(X = x)

Answer

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)?

Accepted Answer

1.5

Answer

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:

Accepted Answer

aE(X) + b

Answer

By linearity of expectation, E(aX + b) = aE(X) + b.

Question 4

Which of the following describes a continuous random variable?

Accepted Answer

Exact time in seconds for a runner to finish a race

Answer

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:

Accepted Answer

E[(X − μ)²]

Answer

Var(X) = E[(X − μ)²] = the expected squared deviation of X from its mean.

AP Practice Test