Chapter 8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables The value of the result of the probability experiment is a RANDOM VARIABLE. If you roll a die we can let X be the number of dots showing, If we have a hand of three cards, X could be the number of clubs in the hand, These examples are all finite discrete random variables. Example Roll a single die and count the number of rolls until a 6 comes up. outcome Y Finally, the random variable can be continuous: Example You are dealt a hand of 5 cards. Find the probability distribution table for the number of hearts. Graph this in a histogram8.2 Expected Value The expected value for the variable X in a probability distribution is = ⋅ + ⋅ ++ ⋅112 2() ( ) ( ) ( )nnEX XPX X PX X PX Example You are dealt a hand of 5 cards. What is the expected number of hearts? Example A sample of mini boxes of raisin bran cereal was selected and the number of raisins in each box was counted. The results are shown in the table below. # of boxes 13 14 15 16 17# of raisins 10 14 19 16 12 Determine the appropriate random variable X and display the data in a probability histogram. What is the expected value of X? What is the MEAN of X? What was the X value that happened the most often? What was the X value that was in the middle? What is the RANGE of X values? Histograms and Averages: The MEAN (expected value) is where the histogram “balances” The MODE is the tallest rectangle. The MEDIAN is where the area is cut in half. The RANGE is the number of rectangles. (remember, some may have a height of 0).Example Find the mean, median and mode of the following test scores: 77, 46, 98, 87, 84, 62, 71, 80, 66, 59, 79, 89, 52, 94, 77, 72, 85, 90, 64, 70 Mean = Median = Mode = Range = Another way to measure spread? QUARTILES 46 52 59 62 64 66 70 71 72 77 77 79 80 84 85 87 89 90 94 98 Q1 = Q3 = IQR = Box and whisker plot Example A game consists of choosing two bills at random from a bag containing 7 one dollar bills and 3 ten dollar bills. The player gets to keep the money picked. How much should be charged to play this game to keep it “fair” (expected value of zero)? Example From a group of 2 women and 5 men a delegation of 2 is chosen. Find the expected number of women in the delegation.ODDS If P(E) is the probability of event E occurring, then the odds in favor of E are () (),()11() ()cPE PEPEPE PE=¹- We usually express the odds as a ratio of whole numbers, :aatob a bb Example If the probability that the Aggies will win a football game is 80%, what are the odds in favor of the Aggies? If we are given the odds we want to be able to find the probability, if the odds in favor of E are given as a:b, then ()aPEab=+ Odds to win the Breeders' Cup Distaff Saturday, November 4th, 2006 Siempre 29/5 Balletto 7/5 Bushfire 18/1 Fleet Indian 2/1 Happy Ticket 21/2 Round Pond 22/1 Spun Sugar 11/1 Summerly30/1 8.3 Variance and Standard Deviation 3 3 3 3 3 2 2 3 4 4 0 0 5 5 5 POPULATION VARIANCE, POPULATION STANDARD DEVIATION,What do we mean by population? This means everyone, so if ALL the members of the population are used to find the mean, we use the symbols m and s. The mean from SAMPLE uses the symbol x. SAMPLE STANDARD DEVIATION, When we have the probability, it is assumed we had the entire population to base it on, so it is appropriate to use m and s. Example Find the mean and standard deviation for the following distribution: Example An exam has an mean of m=75 and a population standard deviation of s=14. What is the probability that a randomly chosen data point is within 1 standard deviation of the mean? What is the probability that a randomly chosen data point is within 2 standard deviations of the mean? 01245 50 55 60 65 70 75 80 85 90 95 10001245 50 55 60 65 70 75 80 85 90 95 100Chebychev's Theorem: For any data distribution with mean m and standard deviation s, the probability that a randomly chosen data point is within kms- ⋅ to kms+ ⋅ is at least 1 – 1/k2. Or, 21()1Pkx kkms ms- ⋅ ££+⋅ ³- Example A probability distribution has a mean of 50 and a standard deviation of 5. a) What is the probability that an outcome of the experiment lies between 35 and 65? b) Find the value of k so that at least 93.75% of the data lies in the range 50-5k to 50+5k. 8.4 The Binomial Distribution In a Bernoulli trial we have the following: - The same experiment repeated several times. - The only possible outcomes of these experiments are success or failure. - The repeated trials are independent so the probability of success remains the same for each trial. Example A multiple choice test has 3 questions, each with 4 possible answers. A student guesses on each question. What is the probability that the student gets exactly two questions correct? BINOMIAL PROBABILITY: If p is the probability of success in a single trial of a binomial (Bernoulli) experiment, the probability of x successes and n-x failures in n independent repeated trials of the same experiment isExample The first half of July was very dry in college station. If each day there was a 20% chance of rain, what is the probability of no rain in the first 15 days in July? DEFINE SUCCESS: n = number of trials = x = number of successes = p = probability of success = binompdf(n, p, x) on the calculator What is the probability of at most 2 rain days? x = number of successes = binomcdf(n, p, x) will give you the sum of the probabilities from 0 to x. For a binomial probability question you must do the following: - Decide that it is a Bernoulli trial. - Define what success is. - Find the number of times the experiment is done, n. - Find the probability of success, p. - Determine the number of successes you need to find, x.Example A new drug being tested causes a serious side effect in 5% of patients. What is the probability that in a sample of 10 patients none get the side effect from taking the drug? define success = no side effect, n = number of trials = x = number of successes = p = probability of success = define success = side effect, n = number of trials = x = number of successes = p =
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