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5.1 Notes Introduction to Random Variables and Probability Distributions Discrete Random Variable – i.e. # Continuous Random Variable – i.e. Ex. 1 Which of the following random variables are discrete and which are continuous? a) Time it takes a student to register for classes b) The number of “bad checks” drawn on a checking account c) The amount of gasoline needed to drive your car 200 miles. d) The amount of voters in the last local election. Probability Distribution – Same as a relative frequency distribution 1. 2. Ex. 2 Dr. Fidget 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, 6, with 6 indicating the highest tolerance for boredom. The test results for this group are shown in the table. a) Find the probability of receiving each score on the boredom test. b) Make a histogram of the results from part a) c) If Top Notch Clothing Company needs to hire someone with a score of 5 or 6 to operate a fabric press machine, what is the probability that a person chosen at random will score 5 or 6 on the test? Boredom Tolerance Test Scores Score # of subjects 0 1400 1 2600 2 3600 3 6000 4 4400 5 1600 6 400 Probability Mean of a probability distribution: Standard Deviation of a probability distribution: Both of the previous values are found more easily by putting x-values in L1 and putting the corresponding probability in L2 and then computing 1 VarStat, L1, L2 Ex. 3 Are we influenced to buy a product because we saw an ad 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: # of Times Buyers Saw Infomercial 1 2 3 4 5* % of Buyers 27% 31% 18% 9% 15% *This category was 5 or more, but will be treated as 5 in this example. a) Is the previous a probability distribution? Justify. b) Find the mean and standard deviation of the distribution. For Continuous Data, use midpoint of the range for then x-values. Assignment p. 178 #2, 3, 6, 8, 11, 12, 13 Linear Combinations of Random Variables Let x1 and x2 be independent random variables with respective means μ1 and μ2, 2 2 and variances 1 and 2 For the linear combination W = ax1 + bx2, the mean, variance, and standard deviation are as follows: μW = W2 W Ex. 3 Let x1 and x2 be independent random variables with respective means μ1 = 75 and μ2 = 50, and standard deviations σ1 = 16 and σ2 = 9. a) Let L = 3 + 2x1. Compute the mean, variance, and standard deviation of L. μL = L2 L b) Let W = x1 + x2. Find the mean, variance, and standard deviation of W. μW = W2 W c) Let W = x1 – x2. Find the mean, variance, and standard deviation of W. μW = W2 W d) Let W = 3x1 – 2x2. Find the mean, variance, and standard deviation of W. μW = W2 W Assignment p.181 #14-16