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AP Statistics Chapter 7 – Random Variables

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

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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)

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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

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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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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

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Law of Large Numbers

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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 X and Y must be independent Any linear combination of independent Normal random variables is also Normally distributed.

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