A probability distribution shows the possible outcomes of an experiment and the probability of each of these outcomes.

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1 Introduction In the previous chapter we discussed the basic concepts of probability and described how the rules of addition and multiplication were used to compute probabilities. In this chapter we expand the study of probability to include the concept of a probability distribution. What is a Probability Distribution? A probability distribution shows the possible outcomes of an experiment and the probability of each of these outcomes. How can we generate a probability distribution? As an example, the possible outcomes on the roll of a single die are shown at right. Probability distribution: A listing of all the outcomes of an experiment and the probability associated with each outcome. Each face should appear on about one-sixth of the rolls. The table shows the possible outcomes and corresponding probabilities for this experiment. It is a discrete distribution because only certain outcomes are possible and the distribution is a result of counting the various outcomes. Number of Spots on Die Probability Fraction Decimal 1 1/6 = /6 = /6 = /6 = /6 = /6 = Total 6/6 = 0002 The following are important characteristics of a discrete probability distribution: The probability of a particular outcome is between 0 and 1 inclusive. The outcomes are mutually exclusive events. The list is exhaustive. So the sum of the probabilities of the various events is equal to This discrete probability distribution, presented above as a table, may also be portrayed in graphic form as shown on the right. By convention the probability is shown on the Y axis (the vertical axis) and the outcomes on the X axis (the horizontal axis). This probability distribution is often referred to as a uniform distribution. A probability distribution can also be expressed in equation form. For example: P(x) = 1/6, where x can assume the values 1, 2, 3, 4, 5, or 6. Random Variable In any experiment of chance, the outcomes occur randomly. A random variable is a value determined by the outcome of an experiment. Random Variable: A quantity resulting from an experiment that, by chance, can assume different values. A random variable may have two forms: discrete or continuous. A discrete random variable may assume only distinct values and is usually the result of counting. Discrete random variable: A variable that can assume only certain clearly separated values. For example, the number of highway deaths in Arkansas on Memorial Day weekend may be 1, 2, 3, Another example is the number of students earning a grade of B in your statistics class. In both instances the number of occurrences results from counting. Note that there can be 12 deaths or 15 B's but there cannot be 163 deaths or B grades.

2 If we measure something, such as the diameter of a tree, the length of a field, or the time it takes to run the Boston Marathon, the variable is called a continuous random variable. Continuous random variable: A variable that can assume one of an infinitely large number of values within certain limitations. In brief, if the problem involves counting something, the resulting distribution is usually a discrete probability distribution. If the distribution is the result of a measurement, then it is usually a continuous probability distribution. What is the difference between a random variable and a probability distribution? A probability distribution lists all the possible outcomes as well as their corresponding probabilities. A random variable lists only the outcomes. We will examine the continuous random variable and the continuous probability distribution in the next chapter. The Mean, Variance, and Standard Deviation of a Probability Distribution In Chapter 3 we computed the mean and variance of a frequency distribution. The mean is a measure of location and the variance is a measure of the spread of the data. In a similar fashion the mean (μ) and the variance (σ 2 ) summarize a probability distribution. The Mean The mean μ, or expected value E(x), is used to represent the central location of a probability distribution. It is also the long-run average value of the random variable. It is computed by the following formula: Mean of a Probability Distribution This formula directs you to multiply each outcome (x) by its probability P(x); and then add the products. Variance and Standard Deviation While the mean is a typical value used to summarize a discrete probability distribution, it does not tell us anything about the spread in the distribution. The variance tells us about the spread or variation in the data. The variance is computed using the following formula: Variance of a Probability Distribution The steps in computing the variance using formula [6-2] are: Subtract the mean (μ ) from each outcome and square these differences. Multiply each squared difference by its probability Sum these products to arrive at the variance. The standard deviation (σ) of a discrete probability distribution is found by calculating the positive square root of σ 2, thus. Binomial Probability Distribution One of the most widely used discrete probability distributions is the binomial probability distribution. Characteristics of a binomial probability experiment: 4. An outcome of an experiment is classified into one of two mutually exclusive categories a success or a failure. The random variable is the number of successes in a fixed number of trials. The probability of a success and failure stays the same for each trial. The trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial. Illustrations of each characteristic are: Each outcome is classified into one of two mutually exclusive categories. An outcome is classified as either a success or a failure. For example, 40 percent of the students at a particular university are enrolled in the College of Business. For a selected student there are only two possible outcomes the student is enrolled in the College of Business (designated a success) or he/she is not enrolled in the College of Business (designated a failure). The binomial distribution is the result of counting the number of successes in a fixed sample size. If we select 5 students, 0, 1, 2, 3, 4, or 5 could be enrolled in the College of Business. This rules out the possibility of 45 of the students being enrolled in the College of Business. That is, there cannot be fractional counts. The probability of a success remains the same from trial to trial. In the example regarding the College of Business, the probability of a success remains at 40 percent for all five students selected. 4. Each sampled item is independent. This means that if the first student selected is enrolled in the College of Business, it has no effect on whether the second or the fourth one selected will be in the College of Business.

3 How a Binomial Probability Distribution is Computed To construct a binomial probability distribution we need to know: (1) The number of trials, designated n. (2) The probability of success (π) on each trial. The binomial probability distribution is constructed using the formula [6-3]: Binomial Probability Distribution denotes a combination of n items selected x at a time n is the number of trials x is the random variable defined as the number of successes π is the probability of success on each trial (Do not confuse it with the mathematical constant 416.) The mean (μ) and variance (σ 2 ) of a binomial distribution can be computed by these formulas. Mean of a binomial distribution Hypergeometric Probability Distribution Variance of a binomial distribution To qualify as a binomial distribution, the probability of a success must remain constant. In some situations this requirement can not be met. This usually happens when the size of the population is small and samples are drawn from the population and not replaced. This causes the probability of a success to change from one trial (or sample) to the next. This means the trials are not independent. As an example, suppose that we know that there are four diet sodas in a cooler containing 12 sodas. The chance of randomly selecting a diet soda from the cooler on the first try is 4/1 However, if a diet soda is selected on the first try and not put back into the cooler (i.e., not replaced), the probability of selecting a diet soda on the second try is 3/1 The probability of selecting a diet soda changed from the first to the second trial the trials are not independent The change in the probability of the event diet soda is selected can only be computed if the size of the population is known. This assumes that the population is finite, that is, that the number of individuals, objects or measurements in the population is relatively small and known. Finite population: A population consisting of a known and relatively small number of individuals, objects, or measurements. An outcome generating process that has all the characteristics of a binomial experiment except that the trials are not independent is called a hypergeometric probability experiment. Characteristics of a hypergeometric probability experiment: An outcome of each trial is characterized into one of two mutually exclusive categories a success or a failure. The random variable is the number of successes in a fixed number of trials. The trials are not independent. 4. We assume that we sample from a finite population without replacement. So, the probability of a success changes from one trial of the experiment to the next. For example, in a beverage cooler with 4 diet sodas and 8 regular sodas, what is the probability of randomly selecting two sodas that are both diet in just two tries? As describes above, the probability of a diet soda on the first try is 4/1 If a diet soda is selected on the first try and not returned to the cooler, the probability of a diet soda on the second trial is 3/1 So, the probability of selecting two diet sodas in just two attempts is (4/12) (3/11) = Note that in order to compute this probability it was necessary to know the size of the population. This probability may also be calculated using the hypergeometric distribution, which is described by the formula: Hypergeometric Distribution N is the size of the population.

4 S is the number of successes in the population. n is the size of the sample or the number of trials. x is the number of successes in the sample. C is the symbol for a combination. In the example N = 12, S = 4, n = 2, and x = Therefore, Hence, the probability of selecting two sodas from a cooler consisting of 4 diet sodas and eight regular sodas and getting diet sodas on both selections is This is the same probability we computed earlier. Poisson Probability Distribution Another discrete probability distribution is the Poisson probability distribution. It describes the number of times some event occurs during a specified interval. The intervals may be time, area, distance, or volume. Characteristics of a Poisson probability experiment: The random variable is the number of times some event occurs during a defined interval. The probability of the event is proportional to the size of the interval. The intervals do not overlap and are independent. The formula for computing the probability of a success is: Poisson Distribution P(x) is the probability for a specified value of x. x is the number of occurrences (successes). μ is the arithmetic mean number of occurrences (successes) in a particular interval. e is the mathematical constant (base of the Napierian logarithm system) The Poisson probability distribution has many applications. For example, it can be used as a model to describe the number of new cars sold by a car salesperson in a week, the number of automobile accidents at a particular intersection in a month, or the number of incidents of a disease in a given area. To demonstrate, suppose that the mean number of new cars sold by a particular car salesperson in a week is 7. Then the probability that this salesperson will sell exactly five new cars next week is: The Poisson probability distribution can also be used to approximate probabilities for a binomial probability distribution when the probability of a success on a single trial is very small and n is large. Note that the mean number of successes, μ, can be determined in binomial situations by nπ, where n is the total number of trials and π is the probability of success. Mean of a Poisson Distribution As an example where the Poisson distribution is applicable, suppose electric utility statements are based on the actual reading of the electric meter. In 1 out of 100 cases the meter is incorrectly read (π = 0.01). Suppose the number of errors that appear in the processing of 500 customer statements approximates the Poisson distribution (n = 500). In this case the mean number of incorrect statements is 5, found by μ = nπ = 500 (0.01). Using formula [6-3], finding the probability of exactly two errors appearing in 500 customer statements is rather tedious. Instead we use formula [6-7] with μ = 5 and x =.

5 Or, we can merely refer to the Poisson distribution in Appendix B.5. Locate by μ = (5.0) at the top of a set of columns. Then find the x of 2 in the left column and read across to the column headed by 5.0. The probability of exactly 2 statement errors is 0.084