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3. Using Table 8.3, determine the following theoretical Poisson probabilities: a. The probability of 0 occurrences when the mean is 0.7 b. The probability of 3 occurrences when the mean is 5 c. The probability of 11 occurrences when the mean is 5 d. The probability of 1 occurrence when the mean is 0.1 4. At a fast-food restaurant, the arrival of customers during the period from 11:00 A.M. to 2:00 P.M. is tracked. The owner is interested in finding out whether or not the arrival of customers follows the Poisson distribution. Go through the three conditions for a Poisson random variable. Which of those three is most clearly being violated in this case? Explain. 5. State the relationship between the theoretical standard deviation and the theoretical mean of a Poisson random variable. What is the theoretical standard deviation in the case of the Prussian army data? 6. The following table shows probabilities obtained in an experiment. The table also has the Number of occurrences Experimental probability Theoretical probability 0 1 2 3 4 5 6 7 0.053 0.153 0.138 0.103 0.225 0.214 0.068 0.047 0.033 0.113 0.193 0.219 0.186 0.126 0.072 0.035 corresponding probabilities under the Poisson distribution with a mean of 3.4. a. Make a histogram of the experimental probabilities, and a histogram of the theoretical probabilities. b. Compare the two histograms. Do the obtained data seem to follow the Poisson distribution with mean 3.4? 7. Reconsider the data given in Exercise 6. Recall that in Chapter 7 we discussed the chi-square test, a test used to determined whether data follow a certain distribution. Assume that in the experiment described Exercise 6, the experimental probabilities were actually based on 1000 observations. a. Reconstruct the results of the experiment, giving the number of outcomes for each of the values. For example, there were 1000 ⫻ 0.053 ⳱ 53 outcomes of 0. b. Using the theoretical probabilities, calculate how many of the 1000 observations are expected for each of the values. For example, we would expect 1000 ⫻ 0.033 ⳱ 33 outcomes of 0. c. Calculate the value of the chi-square statistic for these data. d. What are your conclusions based on the chi-square test? 8. Using Table 8.3, determine the following theoretical Poisson probabilities: a. The probability of either 1, 2, or 3 occurrences when the mean is 0.1. b. The probability of at least 2 occurrences when the mean is 5. c. The probability of at most 3 occurrences when the mean is 1. 8.3 ROLLING A FAIR DIE: THE DISCRETE UNIFORM DISTRIBUTION If you roll a fair six-sided die, as has been done often in this book, you are equally likely to get 1, 2, 3, 4, 5, or 6. There is a name for such a distribution: the discrete uniform distribution from 1 to 6. Discrete means that the possible values are specific values that can be listed. (The opposite of discrete is continuous, which we discuss in Section 8.5 for a uniform distribution.) Uniform means that each of the values is equally likely. Since there are six values in the case of a normal die, the theoretical probability of any one of them appearing is 16 . A 32-sided die, if it is fair, will yield a discrete uniform distribution with outcomes ranging from 1 to 32. It is not necessary to start with 1. For example, a five-sided die could have the numbers ⫺2, ⫺1, 0, 1, 2. Its distribution will be a discrete uniform distribution ranging from ⫺2 to 2. One needs only to specify the lowest and the highest integer. Also, use of a “die” here is unnecessary. A die is merely a tangible physical object to help us imagine generating the data. The essence of the discrete uniform distribution is a sequence of equally spaced integers each having the same theoretical probability. The theoretical mean is just halfway between the low value and the high value: low value Ⳮ high value 2 For example, the theoretical mean for a fair six-sided die is (low valueⳭhigh value)/2 ⳱ (1 Ⳮ 6)/2 ⳱ 3.5. The formula for the theoretical standard deviation is not very intuitive. For completeness, we give it: Theoretical mean ⳱ Theoretical standard deviation ⳱ 冪 (high value ⫺ low value Ⳮ 1)2 ⫺ 1 12 For the six-sided die, this standard deviation is 冪 (6 ⫺ 1 Ⳮ121) ⫺ 1 ⳱ 冪 6 12⫺ 1 36 ⫺ 1 ⳱冪 ⳱ 1.7078 12 Standard deviation ⳱ 2 2 Thus there will be quite a bit of variability from the average of 3.5 in that the typical variation is roughly from 2 to 5. SECTION 8.3 EXERCISES 1. For each of the following discrete uniform distributions, give the value of the theoretical mean and the theoretical standard deviation. a. High value ⳱ 10, low value ⳱ 1 b. High value ⳱ 8, low value ⳱ 4 c. High value ⳱ 5, low value ⳱ ⫺5 d. High value ⳱ 15, low value ⳱ 6 2. Repeat Exercise 1 for these values: a. High value ⳱ ⫺5, low value ⳱ ⫺9 b. High value ⳱ 21, low value ⳱ 10 c. High value ⳱ 0, low value ⳱ ⫺10 d. High value ⳱ 32, low value ⳱ 25 3. You and three of your friends agree to meet for lunch every day. You begin to keep track of how many of your friends beat you to the meeting place each day. You suspect that this random variable has a uniform distribution. a. What are the possible values of the random variable? b. Assuming that this random variable does follow the uniform distribution, what are its theoretical mean and standard deviation? c. The actual data are as follows: Out of a total of 50 meetings . . . There were 10 times when none of your friends were there when you arrived. There were 13 times when one of your friends was there when you arrived. There were 15 times when two of your friends were there when you arrived. There were 12 times when all three of your friends were there when you arrived. Use the chi-square test as described in Chapter 7 to determine whether these observed data do indeed follow the uniform distribution. 8.4 CONTINUOUS DISTRIBUTIONS The binomial and Poisson distributions are both based on counting; the discrete uniform distribution deals with a list of consecutive integers. These are all discrete distributions. It is common to find data based on measurements instead of counts: for example, mountain heights, light bulb lifetimes, incomes, agricultural yields, and so on. In many measurements, we may be able to specify the smallest and largest possible values, but any value in between those, not just integers, may be possible. For example, a lightbulb’s lifetime (the length of time it is continuously on before breaking) cannot be less than 0 hours, although if it is defective it may last exactly 0 hours. There is some maximum length of time—if not 1000 hours, then maybe 10,000 hours or more. But any value between 0 and some large maximum is possible. It may last exactly one hour, or half an hour, but it is more likely to last a less rounded-off time, such as 134.398459987 hours. Of course, in practice one cannot find the exact lifetime, but instead finds it to the nearest minute, second, or maybe 10th of a second. In any case, the actual number of possible values is immense; theoretically, it is infinite if one ignores the fact that all measurements are 1 rounded off (for example, a length might be rounded to the nearest 1000 of an inch). Moreover, measurements are often best viewed as random, in which case it is necessary to develop probability distributions to describe them. For example, we might be randomly sampling measurements of a real population, such as heights of adult females, in which case height can be assigned a continuous probability distribution. Or we might be randomly sampling from a potentially infinitely repeatable experimental process, such as assessing the lifetime of lightbulbs as they come off the production line,