2. 1 7.3 RANDOM VARIABLES When the variables in question are quantitative, they are known as random variables. A random variable, X, is a quantitative variable whose value varies according to the rules of probability. Random variables, like data, can be discrete (integers) or continuous (real numbers).

3. 2 7.3.1 Probability Distribution of a Discrete Random Variable The rules of probability that describe the way a random variable behaves are known as the probability distribution of the random variable. The probability distribution of a random variable, X, written as p(x), gives the probability that the random variable will take on each of its possible values. p(x) = P(X = x) for all possible values of X 7.3 RANDOM VARIABLES

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6. 5 7.3.1.1 Notation and Probabilities We might be interested in finding the probability that the random variable takes on a value that is "at least x," "more than x," "at most x," "less than x," "between X1 and X2;," or "between X1 and X2 inclusive." As an example, we will use a random variable X that can take on values of x = 0, 1, 2, 3,..., n. 7.3 RANDOM VARIABLES

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9. 8 7.3.2 Probability Histograms Random variables and their probability distributions are the models for the populations from which our sample data are taken. A random variable can be displayed with a probability distribution table or a probability distribution histogram. 7.3 RANDOM VARIABLES

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11. 10 7.4.1 The Binomial Model A binomial random variable is the number of success in n trials or in a sample of size n. Certain characteristics define binomial random variables: –There are a fixed number of identical trials of an experiment. –The outcome for each trial of the experiment can be classified in one of two ways: a success, S or a failure, F. 7.3 RANDOM VARIABLES

12. 11 –The probability that a success occurs in any sample element or on any trial of the experiment, , is the same for each element or trial. –The trials of the experiment are independent. –The random variable is the number of successes that occur in the n trials of the experiment. 7.3 RANDOM VARIABLES

13. 12 7.4 THE BINOMIAL PROBABILITY DISTRIBUTION 7.4.2 The Binomial Probability Distribution The random variable X is the number of successes in n trials of the experiment. The probability distribution of X is determined by this formula:

14. 13 7.4.2.1 Binomial Probability Tables There are tables for values of n from 5 to 30. Each table covers a range of values for  from 0.05 to 0.95. A sample of some of the table for n = 5 is shown in Figure 7.1. 7.4 THE BINOMIAL PROBABILITY DISTRIBUTION

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16. 15 7.4.2.1 Binomial Probability Tables 7.4 THE BINOMIAL PROBABILITY DISTRIBUTION

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18. 17 7.4.3 The Mean and Standard Deviation of the BD Since probability distributions are population models, their means and standard deviations are represented by the Greek letters  and . In particular, for a binomial random variable, X, the mean, , and the standard deviation, , are found using the following formulas: 7.4 THE BINOMIAL PROBABILITY DISTRIBUTION