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Section 5.1

This section introduces the concept of random variables and probability distributions. It explains the difference between discrete and continuous random variables and discusses probability distributions and their properties. It also covers expected value and standard deviation, as well as linear combinations of random variables.

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Section 5.1

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  1. Section • 5.1 • Introduction to Random Variables and Probability Distributions

  2. Random Variables • measurements/counts obtains from a statistical experiment • Let x represent the quantitative result • Ex: # of eggs in a robin’s nest or • Ex: daily rainfall in inches

  3. Two Types of RVs • Discrete: things you count, usually whole numbers • Ex: # of students in a statistics class • Continuous: things you measure, usually fractions or decimals • Ex: air pressure of a tire

  4. Probability Distribution • it’s assigning probabilities to each measurement/count • sum of the probabilities must be 1

  5. Ex 1 – Discrete RV • Dr. Mendoza developed a test to measure boredom tolerance. He administered it to a group of 20,000 adults between the ages of 25 and 35. • The possible scores were 0, 1, 2, 3, 4, 5, and 6, with 6 indicating the highest tolerance for boredom. • Find the probabilities for • each score. • b) Are the scores mutually exclusive? • c) A company needs to hire someone with a score on the boredom tolerance test of 5 or 6 to operate the fabric press machine. Find the probability of someone scoring a 5 or 6. • d) Calculate the expected score (mean score).

  6. b. The graph of this distribution is simply a relative-frequency histogram in which the height of the bar over a score represents the probability of that score.

  7. A probability distribution has a mean and standard deviation • The mean is called the “expected value” • The standard deviation is thought of as “risk” • The larger it is the greater the likelihood RV x is different from the mean

  8. Expected Value for discrete RV • Multiply each value times the probability of getting that value. • Add up the products.

  9. Standard Deviation for discrete RV • Subtract each x from the mean. • Square the difference. • Multiply each squared difference by it’s corresponding probability • Add up all the products • Square root the result

  10. Ex: 2 • Are we influenced to buy a product by an ad we saw on TV? • National Infomercial Marketing Association determined the number of times buyers of a product had watched a TV infomercial before purchasing the product. The results are shown here: • Compute the mean and standard deviation of the distribution.

  11. Linear Combinations • Recall: • **Multiplying/adding something to data values, the mean changes by the same amount • **Only multiplying something to data values changes the st. dev.

  12. Ex: 3 • Let x1 and x2 be independent RVs with means 1 = 75 and 2 = 50, and st.dev. 1 = 16 and 2 = 9. • a) Let L = 3 + 2x1. Compute the mean, variance, and st.dev of L.

  13. cont’d • b) Let W = x1 + x2. Find the mean, variance, and st.dev of W.

  14. c) Let W = x1 – x2. Find the mean, variance, and st.dev. of W.

  15. d) Let W = 3x1 – 2x2. Find the mean, variance, and st. dev. of W.

  16. Linear combination formulas

  17. Ex: 4 • A tool of cryptanalysis (science of code breaking) is to use relative frequencies of occurrence of letters to break codes. In addition to cryptanalysis, creation of word games also uses the technique. Oxford Dictionaries publishes dictionaries of English vocabulary. They did an analysis of letter frequencies in words listed in the main entries of the Concise Oxford Dictionary. Suppose someone took a random sample of 1000 words occurring in crossword puzzles. • Find the P(letter will be a vowel):

  18. Ex: 5 • At a carnival, you pay $2.00 to play a coin-flipping game with three fair coins. On each coin one side has the number 0 and the other side has the number 1. You flip the three coins at one time and you win $1.00 for every 1 that appears on top. • What is the random variable? • What is the sample space (all outcomes)? • Calculate the expected earning if you play this game. Is the expected earning less than, equal to, more than the cost of the game?

  19. Section 5.1 Assignment: • pg.205: #1, 3, 7, 8, 11, 13, 15, 17, 19, • 21(omit part d) • You must show your work!

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