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ENGR 224/STAT 224 Probability and Statistics Lecture 8

ENGR 224/STAT 224 Probability and Statistics Lecture 8. Definition: Variance. The Variance of random variable X, having probability mass function, p(x) is denoted as V(x) = s 2. Definition: Shortcut. Rules of Expected Values. Rules of Variance. Practice Problems. Problem 33

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ENGR 224/STAT 224 Probability and Statistics Lecture 8

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  1. ENGR 224/STAT 224 Probability and StatisticsLecture 8

  2. Definition: Variance The Variance of random variable X, having probability mass function, p(x) is denoted as V(x) = s2

  3. Definition: Shortcut

  4. Rules of Expected Values

  5. Rules of Variance

  6. Practice Problems • Problem 33 • Problem 37

  7. Definition: Binomial Experiment A binomial experiment is one that meets the following requirements. • The experiment must have a fixed number of trials • The trials must be independent • Each trial must have all outcomes classified into two categories • The probabilities must remain constant for each trial.

  8. Example: Swervin Mervin Mervin is a basketball player making free throws. The probability of making a basket is 0.6 on each throw. With 5 throws what is the probability that he makes 2 shots.

  9. Notation: • S = success and P(S) = p • F = Failure and P(F) = q = 1- p • n = fixed number of trials • x = specific number of successes in n trials • P(x) = the probability of getting exactly x successes among n trials

  10. Example: Swervin Mervin Mervin is a basketball player making free throws. The probability of making a basket is 0.6 on each throw. With 5 throws what is the probability that he makes 2 shots. For this example we have p = 0.6 probability of success (basket) q = 0.4 probability of failure (miss) n = 5 number of trials x = 2 number of successes in 3 trials P(2) probability of making 2 shots

  11. Binomial Probability Distribution In a binomial experiment, with constant probability p of success at each trial, the probability of x successes in n trials is given by

  12. Example: Swervin Mervin Mervin is a basketball player making free throws. The probability of making a basket is 0.6 on each throw. With 5 throws what is the probability that he makes 2 shots. Solution: Find the probabilities of Marvin making 0, 1 and 3 baskets. P(0)=.010 P(1)=.077 P(2)= .230 P(3) = .346

  13. How to use the Binomial Tables • (see page 664) • First find the appropriate table for the particular value of n • then find the value of p in the top row • Find the row corresponding to k and find the intersection with the column corresponding to the value of p • The value you obtain is the cumulative probability, that is P(x ≤ k) • N=5, p = 0.6: P(x = 2) = 0.317-0.087=0.230 • N=5, p = 0.6: P(x = 3) = 0.663 - 0.317 = 0.346 • If we want the probability of at least one basket, then • N=5, p = 0.6: P(x ≥ 1) = 1 - P(x ≤ 0) =1-0.010=0.990

  14. Example: Flipping a biased Suppose that 5% of all fuses are known to be defective. In a sample of size 20, what is the probability of selecting exactly 1 defective fuse and the probability of selecting at least 1 defective fuse. Solution:

  15. Mean and Standard deviation of a binomial distribution

  16. Practice Exercises • Problem 51 • Problem 55

  17. Overview • Binomial Experiments • Mean or Expected Value, • Standard Deviation

  18. Homework • For next class reread chapters 1 thru 3 • Try Problem 106 page 127 • Read Sections 4.1 and 4.2

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