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Section 6.2 Part #1: Transforming Random Variables AP Statistics Name: ___________________ Date: ___________ Period: __ Scenario: Peteβs Jeep Tours offers a popular half-day trip in a tourist area. There must be at least 2 passengers for the trip to run, and the vehicle can hold up to 6 passengers. The number of passengers X on a randomly selected day has the following probability distribution. No. of passengers xi Probability pi 2 3 4 5 6 0.15 0.25 0.35 0.20 0.05 a) Draw a histogram of the probability distribution of X. Describe the shape of the histogram. b) Find the mean of X and interpret it. Then find the standard deviation of X and interpret it. ππ₯ = πΈ(π) = __________ ππ₯2 = __________ ππ₯ = __________ Scenario #2: Pete charges $150 per passenger. Let C = the total amount of money that Pete collects on a randomly selected trip. Because the amount of money Pete collects from the trip is just $150 times the number of passengers, we can write the equation C = 150X. From the probability distribution of X, we can see that the chance of having two people (X = 2) on the trip is 0.15. In that case, C = 150(2) = 300. So one possible value of C is $300, and its corresponding probability is 0.15. If X = 3, then C = 150(3) = 450, and the corresponding probability is 0.25. Fill in the remaining boxes of probability distribution of C. Total Collected ci 300 450 Probability pi 0.15 0.25 0.35 0.20 0.05 a) Draw a histogram of the probability distribution of C. Describe the shape. b) Find the mean of C and the standard deviation of C. ππ = πΈ(πΆ) = __________ ππ2 = _________ ππ = __________ Summarize: Multiplying/Dividing a Random Variable by b οΌ It will multiply/divide the mean by ________ οΌ It will multiply/divide the standard deviation by ________ οΌ It will multiply/divide the variance by __________ οΌ The shape of the distribution will _______________________________. οΌ Can you recall what it will do to the median and quartiles? Can you recall what it will do to the range? (Think back to Chapter 2) Scenario #3: It costs Pete $100 to buy permits, gas, and a ferry pass for each half-day trip. The amount of profit V that Pete makes from the trip is the total amount of money C that he collects from the passengers minus $100. That is, V = C β 100. If Pete has only two passengers on the trip (X = 2), then C = 300 and V = $200. From the probability distribution of C, the chance that this happens is 0.15. So the smallest possible value of V is $200; its corresponding probability is 0.15. If X = 3, then C = 450 and V = 350, and the corresponding probability is 0.25. Fill in the remaining boxes of probability distribution of V. Total Collected vi 200 350 Probability pi 0.15 0.25 0.35 0.20 0.05 a) Draw a histogram of the probability distribution of V. Describe the shape. b) Find the mean of V and the standard deviation of V. ππ£ = πΈ(π) = __________ ππ£2 = _________ ππ£ = __________ Summarize: Adding/Subtracting a Random Variable by a οΌ It will add/subtract ______ to the mean. οΌ How will it affect the variance and standard deviation? οΌ The shape of the distribution will _______________________________. οΌ Can you recall what it will do to the median and quartiles? Can you recall what it will do to the range? (Think back to Chapter 2)