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Chapter 12 Probability and Calculus

Chapter 12 Probability and Calculus. Chapter Outline. Discrete Random Variables Continuous Random Variables Expected Value and Variance Exponential and Normal Random Variables Poisson and Geometric Random Variables. § 12.1. Discrete Random Variables. Section Outline. Mean Expected Value

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Chapter 12 Probability and Calculus

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  1. Chapter 12Probability and Calculus

  2. Chapter Outline • Discrete Random Variables • Continuous Random Variables • Expected Value and Variance • Exponential and Normal Random Variables • Poisson and Geometric Random Variables

  3. §12.1 Discrete Random Variables

  4. Section Outline • Mean • Expected Value • Variance • Standard Deviation • Frequency Table • Relative Frequency Table • Relative Frequency Histogram • Random Variable • Applications of Expected Value, Variance, and Standard Deviation

  5. Mean

  6. Expected Value • a1 is the first number from a set of numbers, a2 is the second and so on • p1 is the probability that a1 occurs, p2 is the probability that a2 occurs and so on

  7. Variance • m is the expected value (or mean) of the set of numbers • a1 is the first number from a set of numbers, a2 is the second and so on • p1 is the probability that a1 occurs, p2 is the probability that a2 occurs and so on

  8. Standard Deviation

  9. Frequency Table (Distribution)

  10. Relative Frequency Table (Probability Table)

  11. Relative Frequency Histogram

  12. Random Variable

  13. Applications of Expected Value, Variance, & Standard Deviation EXAMPLE Find E(X), Var (X), and the standard deviation of X, where X is the random variable whose probability table is given in Table 5. SOLUTION

  14. Applications of Expected Value EXAMPLE The number of phone calls coming into a telephone switchboard during each minute was recorded during an entire hour. During 30 of the 1-minute intervals there were no calls, during 20 intervals there was one call, and during 10 intervals there were two calls. A 1-minute interval is to be selected at random and the number of calls noted. Let X be the outcome. Then X is a random variable taking on the values 0, 1, and 2. (a) Write out a probability table for X. (b) Compute E(X). (c) Interpret E(X). SOLUTION (a)

  15. Applications of Expected Value CONTINUED (b) (c) Since E(X) = 0.67, this means that the expected number of phone calls in a 1-minute period is 0.67 phone calls.

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