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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.
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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: x1 x2 x3 x4 . . . .xk • P(X) : p1 p2 p3 p4. . . .pk
Example of a Probability Distribution (Discrete RV) • Xage when male college students began to shave regularly. • X 11 12 13 14 15 16 17 18 19 20 • p(x).013 0 .027 .067 .213 .267 .240 .093 .067 .013
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 μx, μy, 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. • μx = ∑ xi pi X: x1 x2 x3 x4 . . . . xk P(X): p1 p2 p3 p4. . . . pk
Example of a Probability Distribution (Discrete RV) • Xage when male college students began to shave regularly. • X 11 12 13 14 15 16 17 18 19 20 • p(x).013 0 .027 .067 .213 .267 .240 .093 .067 .013
Variance/ Standard Deviation • The variance of a random variable is represented by σ2x and the standard deviation by σx. • For a discrete random variable… • σ2x = ∑(xi – μx)2 pi
Law of Large Numbers • As the sample size increases, the sample mean approaches the population mean.
Rules for means of Random Variables • 1.μa+bx = a + bμx • If you perform a linear transformation on every data point, the mean will change according to the same formula. • 2. μX ± Y = μX ± μY • 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. σ2a + bx = b2σ2x • 2. σ2X + Y = σ2X + σ2Y • σ2X - Y = σ2X + σ2Y • X and Y must be independent • Any linear combination of independent Normal random variables is also Normally distributed.