Bayes s Rule Example. defective. An MP3 player is selected at random and found to be defective. What is the probability it came from Factory I?

Transcription

1 Bayes s Rule Example A company manufactures MP3 players at two factories. Factory I produces 60% of the MP3 players and Factory II produces 40%. Two percent of the MP3 players produced at Factory I are defective, while 1% of Factory II s are defective. An MP3 player is selected at random and found to be defective. What is the probability it came from Factory I?

2 Chapter 16 Random Variables

3 Random Variable A random variable is a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point. We use a capital letter, like X, to denote a random variable. A particular value of a random variable will be denoted with a lower case letter, in this case x.

4 Random Variable (cont.) There are two types of random variables: Discrete random variables can take one of a finite number of distinct outcomes. Example: Number of credit hours Continuous random variables can take any numeric value within a range of values. Example: Cost of books this term

5 Discrete Random Variable Examples Experiment Random Variable Possible Values Make 100 Sales Calls # Sales 0, 1, 2,..., 100 Inspect 70 Radios # Defective 0, 1, 2,..., 70 Answer 33 Questions # Correct 0, 1, 2,..., 33 Count Cars at Toll Between 11:00 & 1:00 # Cars Arriving 0, 1, 2,...,

6 Continuous Random Variable Examples Experiment Random Variable Possible Values Weigh 100 People Weight 45.1, 78,... Measure Part Life Hours 900, 875.9,... Amount spent on food $ amount 54.12, 42,... Measure Time Between Arrivals Inter-Arrival Time 0, 1.3, 2.78,...

7 Thinking Challenge Which of the following describe continuous random variables? Which describe discrete random variables? a)the number of newspapers sold by the New York Times each month b)the amount of ink used in printing a Sunday edition of the New York Times. c)the actual number of ounces in a 1-gallon bottle of laundry detergent. d)the number of defective parts in ashipment of nuts and bolts. e)the number of people collecting unemployment insurance each month.

8 Probability model (distribution) A probability model for a random variable consists of: The collection of all possible values of a random variable, and the probabilities that the values occur.

9 Discrete Probability Distribution The probability distribution of a discrete random variable is a graph, table, or formula that specifies the probability associated with each possible value the random variable can assume.

10 Requirements for the Probability Distribution of a Discrete Random Variable x 1. p(x) 0 for all values of x 2. Σp(x) = 1 where the summation of p(x) is over all possible values of x.

11 Discrete Probability Distribution Example Experiment: Toss 2 coins. Count number of tails. Probability Distribution Values, x Probabilities, p(x) 0 1/4 = /4 = /4 = T/Maker Co.

12 Visualizing Discrete Probability Distributions p(x) Graph x # Tails Table f(x) Count p(x) Formula n! P(x) = p x!(n x)! x (1 p) n x

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15 Thinking challenge A fair coin is tossed till to get the first head or four tails in a row. Let X be the number of tails we get. a. Determine the values of random variables b. Assign probabilities to each value of random b. Assign probabilities to each value of random variable

16 Expected Value and Variance Of particular interest is the value we expect a random variable to take on, notated µ (for population mean) or E(X) for expected value. Expected Value represents the center of the probability distribution. Variance, σ 2, is for the dispersion of the probability distribution. Standard deviation is the square root of the variance.

17 Ex You play two games against the same opponent. The probability you win the first game is 0.4. If you win the first game, the probability you also win the second game is 0.2. If you lose the fırst game, the probability that you win the second is 0.3 Are the two games independent? What is the probability that you lose the both games? What is the probability that you win the both games? If you lost the second game, what is the probability that you won the first game? Let random variable X be the number of games you win. Find the probability dist.(model) for X.

18 Continuous Random Variables Random variables that can take on any value in a range of values are called continuous random variables. Continuous random variables have means (expected values) and variances. We won t worry about how to calculate these means and variances in this course, but we can still work with models for continuous random variables when we re given these parameters.

19 Thinking Challenge Your company bids for two contracts. You believe the probability you get contract#1 is 0.8. If you get contract #1, the probability you will also get contract#2 will be 0.2, and if you do not get #1, the probability you get #2 will be 0.3. a) Are the two contracts independent? Explain. b) Find the probability you get both contracts. c) Find the probability you get no contract. d) If you did not get the contract#2, what is the probability that you got contract#1? e) Let X be the number of contracts you get. Find the probability model for X.