AP Statistics Chapter 7 Notes

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

Example of a Probability Distribution (Discrete RV) X  age when male college students began to shave regularly. 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. –The “expected value” does not have to be a number that can possibly be obtained, therefore you can’t necessarily “expect” it to occur.

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

Example of a Probability Distribution (Discrete RV) X  age when male college students began to shave regularly. X p(x)

Variance/ Standard Deviation The variance of a random variable is represented by σ 2 x and the standard deviation by σ x. For a discrete random variable… σ 2 x = ∑(x i – μ x ) 2 p i

Law of Large Numbers As the sample size increases, the sample mean approaches the population mean.

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 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.