AP Statistics

Chapter 7 – Random Variables

Random Variables

• • Random Variable – A variable whose value is a numerical outcome of a random phenomenon.

Discrete Random Variable – Has a countable number of outcomes – e.g. Number of boys in a family with 3 children • (0, 1, 2, or 3)

Probability Distribution

• Lists the values of a discrete random variable and their probabilities.

Value of X: P(X) : x 1 p 1 x 2 p 2 x 3 p 3 x 4 . . . . x k p 4 . . . . p k

Example of a Probability Distribution (Discrete RV) • X  RV that counts the number of “Tails” in three tosses of a balanced coin. • • X p(x) ______________________________________

Continuous Random Variable

• • Takes on all values in an interval of numbers.

– e.g. women’s heights – e.g. arm length Probability Distribution for Continuous RV – Described by a density curve.

– The probability of an event is the area under a density curve for a given interval.

– e.g. a Normal Distribution

• •

Mean

The mean of a random variable is represented by μ 𝑋 , μ 𝑌 , etc.

The mean of X is often called the expected value of X.

•

Mean Formula

For a discrete random variable with the distribution.

X: P(X): x1 p1 • μ x = ∑ (x i * p i ) x2 p2 x3 p3 x4 . . . . xk p4. . . . pk

• • •

Variance/ Standard Deviation

The variance of a random variable is represented by σ 2 x The standard deviation of a random variable is represented by σ x .

For a discrete random variable: X: P(X): x1 p1 x2 p2 x3 p3 x4 . . . . xk p4. . . . pk • σ 2 x = ∑(x i – μ x ) 2 p i

• •

Law of Large Numbers

As the sample size increases, the sample mean approaches the ________________________.

Holds true for _________________________.

– Draw independent observations at random from any population with finite mean μ.

– Decide how accurately you would like to estimate the mean μ.

– As the number of observations drawn increases, the mean 𝑥 of the observed values eventually approaches the mean μ of the population as closely as you specified and then stays that close.

Rules for means of Random Variables 1.μ

a+bx –

= a + bμ

x If you perform a linear transformation on every data point, the mean will change according to the same formula.

2. μ

X + Y –

= μ

X

+ μ

Y If you combine two variables into one distribution by adding or subtracting, the mean of the new distribution can be calculated using the same operation.

Rules for variances of Random Variables 1. σ 2 a + bx = b 2 σ 2 x 2. σ 2 X + Y = σ 2 X + σ 2 Y σ 2 X - Y = σ 2 X + σ 2 Y • X and Y must be independent • Any linear combination of independent Normal random variables is also Normally distributed.