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Statistics and Probability Lecture Notes # Content: Random Variables and Probability Distributions Competencies 4. Are you a coffee drinker? If so, how many 5. Illustrates the mean and variance of a discrete random coffee breaks do you take when you are at variable 6. Calculates the mean and the variance of a discrete work or at school? Most coffee drinkers take a random variable little time for their favorite beverage, and 7. Interprets the mean and the variance of a discrete many take more than one coffee break every random variable day. The table below, adapted from a 8. Solves problems involving mean and variance of Snapshot in USA Today shows the probability probability distributions distribution for x, the number of daily coffee breaks taken per day by coffee drinkers. Problem Solving X 0 1 2 3 4 5 1. At a raffle, 1500 tickets are sold at P50 each P(x) .28 .37 .17 .12 .05 .01 for four prizes P5000, P2500, P1500, and P750. You buy one ticket. What is the expected value of your gain? 5. Your father said, if in one semester your general average is 90 and above, he will add 2. An electronics store sells a particular model of 100 Php to your daily allowance, 50 Php if computer notebook. There are only four your average is 85 to 89, but decrease it by 20 notebooks in stock, and the manager wonders Php if your grade is 84 and below. If the what today's demand for this particular model probability to get 90 and above is 18.5%, will be. She learns from the marketing while you have 43% chance to get 85 to 89, department that the probability distribution what is your expected allowance per for x, the daily demand for the laptop, is as semester? What is the standard deviation of shown in the table. Find the mean, variance, your possible allowance? and standard deviation of x. X 0 1 2 3 4 5 P(x) .10 .40 .20 15 .10 .05 6. A publisher introduces a weekly magazine that sells for Php 395.00. The company 3. The maximum patent life for a new drug is 17 marketers estimate that sales x (in thousand) years. Subtracting the length of time required will be approximated by the following by the FDA for testing and approval of the probability distribution. drug provides the actual patent life of the x 10 15 20 25 30 35 drug-that is, the length of time that a company has to recover research and development costs and make a profit. Suppose the distribution of the lengths of P(x) patent life for new drugs is as shown here: 0.200 0.300 0.025 0.250 0.075 Years, 3 8 0.150 X P(x) .03 05 .07 10 .14 .20 Find the mean, standard deviation, and Years, 9 10 11 12 13 expected value of the probability distribution. X Interpret the results in the context of the real- P(x) 18 .12 .07 .03 .01 life situation. a. Find the expected number of years of patent life for a new drug. b. Find the standard deviation of x