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Chapter 7 Random Variables YMS – 7.1

Chapter 7 Random Variables YMS – 7.1. Discrete and Continuous Random Variables. Find the mean of the following list: 2, 2, 2, 6, 8. Vocabulary. Random variable One whose value is a numerical outcome of a random phenomenon Probability Histograms Same as relative frequency histograms

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Chapter 7 Random Variables YMS – 7.1

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  1. Chapter 7Random VariablesYMS – 7.1 Discrete and Continuous Random Variables

  2. Find the mean of the following list: 2, 2, 2, 6, 8

  3. Vocabulary • Random variable • One whose value is a numerical outcome of a random phenomenon • Probability Histograms • Same as relative frequency histograms • Discrete random variable • Has a countable number of possible values • Every probability pi is a number between 0 and 1

  4. Continuous probability distributions • Takes all values in an interval of numbers • Assign zero to every individual outcome • Makes no distinction between ≤ and < (Read the bottom of p399) • Normal Distributions • Z-scores • Standard Normal Distribution • N(µ, σ) p396 #7.2, 7.7-7.8, 7.18-7.19

  5. YMS 7.2 Mean and Variances of Random Variables

  6. Mean of RVs • Weighted averages • Long-run average after a large number of trials • Sometimes called expected value of X • Denoted by μx • Multiply each possible value by its probability and then add all the products

  7. Variance of RVs • Measure of spread that accompanies mean • Is an average of the squared deviation of the variable X from its mean μx • Denoted by σx2 • Find squared deviation from the mean and multiply by corresponding probability, then add

  8. Law of Large Numbers • Draw independent observations at random • Finite mean μ • Explains why it is a reasonable estimate of μ even though it varies from sample to sample • Variability • The more variable the outcome, the more trials are needed to ensure that is close to μ p412 #7.24 – 7.25

  9. Rules • Any linear combo of independent normal random variables is also normally distributed • Means • If X is a random variable and a and b are fixed numbers, then µa+bX = a + bµX • If X and Y are random variables, then µX±Y = µx ± µY

  10. Variances • If X is a random variable and a and b are fixed numbers, then • If X and Y have correlation ρ, then • If variables are independent, ρ = 0 • Standard Deviations do not add!

  11. 7.2 Homework 7.24-7.25 7.32 7.37-7.38 7.41 Ch 7 Review 7.55-7.58 7.61 7.64-7.67

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