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AP Statistics Chapter 7 Notes

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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)

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Probability Distribution Lists the values of a discrete random variable and their probabilities. Value of X: x 1 x 2 x 3 x 4.... x k P(X) :p 1 p 2 p 3 p 4.... p k P(X) :p 1 p 2 p 3 p 4.... p k

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Example of a Probability Distribution (Discrete RV) X age when male college students began to shave regularly. X11 12 13 14 15 16 17 18 19 20 p(x).013 0.027.067.213.267.240.093.067.013

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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

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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.

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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

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Example of a Probability Distribution (Discrete RV) X age when male college students began to shave regularly. X11 12 13 14 15 16 17 18 19 20 p(x).013 0.027.067.213.267.240.093.067.013

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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

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

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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.

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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.

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Probability and inference Random variables IPS chapters 4.3 and 4.4 © 2006 W.H. Freeman and Company.

Probability and inference Random variables IPS chapters 4.3 and 4.4 © 2006 W.H. Freeman and Company.

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