Copyright © Cengage Learning. All rights reserved. 8 PROBABILITY DISTRIBUTIONS AND STATISTICS

Copyright © Cengage Learning. All rights reserved. 8.1 Distributions of Random Variables

3 Probability Distributions of Random Variables

4 Probability Distributions of Random Variables Since the random variable associated with an experiment is related to the outcomes of the experiment, it is clear that we should be able to construct a probability distribution associated with the random variable rather than one associated with the outcomes of the experiment. Such a distribution is called the probability distribution of a random variable and may be given in the form of a formula or displayed in a table that gives the distinct (numerical) values of the random variable X and the probabilities associated with these values.

5 Probability Distributions of Random Variables Thus, if x 1, x 2,..., x n are the values assumed by the random variable X with associated probabilities P(X = x 1 ), P(X = x 2 ),...,P(X = x n ), respectively, then the required probability distribution of the random variable X may be expressed in the form of the table shown in Table 3, where p i = P(X = x i ), i = 1, 2,..., n. Table 3

6 Probability Distributions of Random Variables The probability distribution of a random variable X satisfies 1. 0  p i  1 i = 1, 2,..., n 2. p 1 + p 2 + · · · + p n = 1 In the next example, we illustrates the construction and application of probability distributions.

7 Applied Example 6 – Waiting Lines The following data give the number of cars observed waiting in line at the beginning of 2-minute intervals between 3 P.M. and 5 P.M. on a certain Friday at the drive-in teller of Westwood Savings Bank and the corresponding frequency of occurrence.

8 Applied Example 6 – Waiting Lines a. Find the probability distribution of the random variable X, where X denotes the number of cars observed waiting in line. b. What is the probability that the number of cars observed waiting in line in any 2-minute interval between 3 P.M. and 5 P.M. on a Friday is less than or equal to 3? Between 2 and 4, inclusive? Greater than 6? cont’d

9 Applied Example 6(a) – Solution Dividing each number in the second row of the given table by 60 (the sum of these numbers) gives the respective probabilities associated with the random variable X when X assumes the values 0, 1, 2,..., 8. (Here, we use the relative frequency interpretation of probability). For example, P(X = 0) = .03

10 Applied Example 6(a) – Solution P(X = 1) = .15 and so on. The resulting probability distribution is shown in Table 6. cont’d Table 6

11 Applied Example 6(b) – Solution The probability that the number of cars observed waiting in line is less than or equal to 3 is given by P(X  3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = =.65 The probability that the number of cars observed waiting in line is between 2 and 4, inclusive, is given by P(2  X  4) = P(2) + P(3) + P(4) = =.60 cont’d

12 Applied Example 6(b) – Solution The probability that the number of cars observed waiting in line is greater than 6 is given by P(X > 6) = P(7) + P(8) = =.05 cont’d

13 Histograms

14 Histograms A probability distribution of a random variable may be exhibited graphically by means of a histogram. To construct a histogram of a particular probability distribution, first locate the values of the random variable on a number line. Then, above each such number, erect a rectangle with width 1 and height equal to the probability associated with that value of the random variable.

15 Histograms For example, the histogram of the probability distribution appearing in Table 4 is shown in Figure 1. Figure 1 Histogram showing the probability distribution for the number of heads occurring in three coin tosses Table 4

16 Histograms Observe that in histogram, the area of a rectangle associated with a value of a random variable X gives precisely the probability associated with the value of X. This follows because each such rectangle, by construction, has width 1 and height corresponding to the probability associated with the value of the random variable. Another consequence arising from the method of construction of a histogram is that the probability associated with more than one value of the random variable X is given by the sum of the areas of the rectangles associated with those values of X.

17 Histograms For example, in the coin-tossing experiment of Example 1, the event of obtaining at least two heads, which corresponds to the event (X = 2) or (X = 3), is given by P(X = 2) + P(X = 3) and may be obtained from the histogram depicted in Figure 1 by adding the areas associated with the values 2 and 3 of the random variable X. Figure 1 Histogram showing the probability distribution for the number of heads occurring in three coin tosses

18 Histograms We obtain P(X = 2) + P(X = 3) = This result provides us with a method of computing the probabilities of events directly from the knowledge of a histogram of the probability distribution of the random variable associated with the experiment.

19 Example 7 Suppose the probability distribution of a random variable X is represented by the histogram shown in Figure 4. Identify the part of the histogram whose area gives the probability P(10  X  20). Figure 4

20 Example 7 – Solution The event (10  X  20) is the event consisting of outcomes related to the values 10, 11, 12,..., 20 of the random variable X. The probability of this event P(10  X  20) is therefore given by the shaded area of the histogram in Figure 5. Figure 5 P(10  X  20)

21 Practice p. 435 Exercises #24 p. 433 Self-Check Exercises #2

22 Assignment p. 434 Exercises #13-19, 25-29