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Chapter 5

Chapter 5. This chapter will deal with the construction of probability distributions by combining the methods of Chapter 3 with the those of Chapter 4. Probability Distributions will describe what will probably happen instead of what actually did happen.

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Chapter 5

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  1. Chapter 5 This chapter will deal with the construction of probability distributions by combining the methods of Chapter 3 with the those of Chapter 4. Probability Distributions will describe what will probably happen instead of what actually did happen.

  2. Combining Descriptive Statistics Methods and Probabilities to Form a Theoretical Model of Behavior

  3. Definitions • Probability Distribution • Random Variable • Discrete random variable Continuous random variable

  4. Example 1: Probability DistributionNumber of Girls Among three Newborn Babies GGG BBB GBG GGB BGG GBB BGB BBG x P(x) 0 1 2 3 1/8 3/8 3/8 1/8 The (discrete) random variable x represents the number of girls born

  5. A probability distribution predicts what will happen. The actual results, however, can differ because of random chance.

  6. Requirements for Probability Distribution

  7. Requirements for Probability Distribution  P(x) = 1 where x assumes all possible values

  8. Requirements for Probability Distribution  P(x) = 1 where x assumes all possible values 0 P(x) 1 for every value of x

  9. Example: 2 Which of the following is a probability distribution?

  10. Mean, Variance and Standard Deviation of a Probability Distribution µ = [x•P(x)] = [x 2 • P(x)] - µ2 The variance is found by squaring the standard deviation Find the mean and standard deviation for example 1

  11. Find the mean and standard deviation for example 1 • See page 169 for TI-83 instructions • Enter the values of the random variable x in list L1. Enter the corresponding probabilities in list L2. Press STAT, select CALC, then select 1-VAR STATS. Enter L1, L2 (with the comma!) then press the ENTER key. µ = 1.5  = 0.9

  12. Definition Expected Value The average value of outcomes E =  [x • P(x)]

  13. E =  [x • P(x)] Example 3: The payoff for “pick three” is 499 to 1. This means that for each winning $1 bet, you would be given $500, a net return of $499 suppose that you bet a $ 1 what is your expected value of gain or loss? P(x) 0.001 0.999 Event Win Lose x $499 - $1

  14. E =  [x • P(x)] Event Win Lose x $499 - $1 P(x) 0.001 0.999 x • P(x) 0.499 - 0.999

  15. E =  [x • P(x)] Event Win Lose x $499 - $1 P(x) 0.001 0.999 x • P(x) 0.499 - 0.999 E = -$.50

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