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: x 1 x 2 x 3 x x k P(X) :p 1 p 2 p 3 p 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

Mean Formula For a discrete random variable with the distribution. μ x = ∑ (x i * p i ) X:x1x2 x3 x4.... xk X:x1x2 x3 x4.... xk P(X):p1 p2 p3 p4.... pk P(X):p1 p2 p3 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: σ 2 x = ∑(x i – μ x ) 2 p i X:x1x2 x3 x4.... xk X:x1x2 x3 x4.... xk P(X):p1 p2 p3 p4.... pk P(X):p1 p2 p3 p4.... pk

Law of Large Numbers

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.