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After this section, you will be able to: 1. Distinguish between discrete and continuous random variables. 2. Graph discrete probability distributions. 3. Compute 𝜇 and 𝜎 for a discrete probability distribution. 4. Compute 𝜇 and 𝜎 for a linear function of a random variable x. 5. Compute 𝜇 and 𝜎 for a linear combination of two independent random variables. ______________: a variable (typically x) that has a single _________________, determined by ________________, for each outcome of a procedure. _____________ random variable: can take on only a ___________ number of values or a ___________________ number of values. “________________” refers to the fact that there might be ____________________ many values, but they result from the counting process. _____________ random variable: can take on any of the countless number of values in a __________________________________. There are no gaps or interruptions. Which of the following random variables are discrete and which are continuous? 1. Let x = the number of eggs that a hen lays in a day. 2. The count of the number of patrons attending a rock concert. 3. Let x = the amount of milk a cow produces in a day. 4. The measure of voltage for a smoke-detector battery. x (girls) P(x) 0 0.000 1 0.001 2 0.006 _________________, or ___________________ that gives the 3 0.022 probability for each value of the random variable. 4 0.061 5 0.122 6 0.183 7 0.209 8 0.183 9 0.122 10 0.061 11 0.022 12 0.006 13 0.001 14 0.000 A probability distribution is a ________________, Features: 1. The probability distribution has a probability assigned to ______________ distinct value of the random variable. 2. The sum of all the assigned probabilities must be ______. 0.25 Probability 0.2 Probability Histogram 0.15 0.1 0.05 0 1 2 3 4 5 6 7 8 9 10 11 12 Number of Girls among Fourteen Newborns 13 14 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. Here are the results: Boredom Tolerance Test Scores 20,000 subjects Score Number of Subjects 0 1400 1 2600 2 3600 3 6000 4 4400 5 1600 6 400 Probability Distribution of Scores on Boredom Tolerance Test Score x 0 1 2 3 4 5 6 Probability P(x) continued b. Draw a relative-frequency histogram for the test scores. c. The Topnotch Clothing Company needs to hire someone with a score on the boredom tolerance test of 5 or 6 to operate the fabric press machine. What is the probability that someone will be randomly chosen from the group who took the test and scored either 5 or 6? Referring to the probability distribution of a __________________, we use: _________ for the mean ________ = 𝑥𝑃(𝑥) _______ for the standard deviation ________ = (𝑥 − 𝜇)2 𝑃(𝑥) • X is the value of the random variable • P(x) is the probability of that variable • The sum is taken for all values of the random variable. 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 watched a TV infomercial before purchasing the product. The results are as follows: Number of Times Buyers Saw Infomercial Percentage of Buyers 1 2 3 4 5 27% 31% 18% 9% 15% We can treat the information shown as an estimate of the probability distribution because he events are mutually exclusive and the sum of the percentages is 100%. Compute the mean and standard deviation of the distribution. continued Number of Times Buyers Saw Infomercial Percentage of Buyers 𝜇= 𝑥𝑃(𝑥) 1 2 3 4 5 27% 31% 18% 9% 15% 𝜎= (𝑥 − 𝜇)2 𝑃(𝑥) Number of Times Buyers View Infomercial Before Making Purchase x P(x) 1 2 3 0.27 0.31 0.18 4 5 0.09 0.15 Page 178-179 #s: 1 – 8 (skip #5)