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1. Frequency Distribution & Relative Frequency Distribution 2. Histogram of Probability Distribution 3. Probability of an Event in Histogram 4. Random Variable 5. Probability Distribution of a Random Variable 1

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2 A table that includes every possible value of a statistical variable with its number of occurrences is called a frequency distribution. If instead of recording the number of occurrences, the proportion of occurrences are recorded, the table is called a relative frequency distribution.

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3 Two car dealerships provided a potential buyer sales data. Dealership A provided 1 year's worth of data and dealership B, 2 years' worth. Convert the following data to a relative frequency distribution.

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5 Put the relative frequency distributions of the previous example into a histogram. Note: The area of each rectangle equals the relative frequency for the data point.

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The histogram for a probability distribution is constructed in the same way as the histogram for a relative frequency distribution. Each outcome is represented on the number line, and above each outcome we erect a rectangle of width 1 and of height equal to the probability corresponding to that outcome. 6

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Construct the histogram of the probability distribution for the experiment in which a coin is tossed five times and the number of occurrences of heads is recorded. 7

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In a histogram of a probability distribution, the probability of an event E is the sum of the areas of the rectangles corresponding to the outcomes in E. 9

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For the previous example, shade in the area that corresponds to the event "at least 3 heads." Area is shaded in blue. 10

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Consider a theoretical experiment with numerical outcomes. Denote the outcome of the experiment by the letter X. Since the values of X are determined by the unpredictable random outcomes of the experiment, X is called a random variable. 11

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If k is one of the possible outcomes of the experiment, then we denote the probability of the outcome k by Pr( X = k ). The probability distribution of X is a table listing the various values of X and their associated probabilities p i with p 1 + p 2 + … + p r = 1. 12

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Consider an urn with 8 white balls and 2 green balls. A sample of three balls is chosen at random from the urn. Let X denote the number of green balls in the sample. Find the probability distribution of X. 13

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There are equally likely outcomes. X can be 0, 1, or 2. 14

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Let X denote the random variable defined as the sum of the upper faces appearing when two dice are thrown. Determine the probability distribution of X and draw its histogram. 15

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X = 3 The sample space is composed of 36 equally likely pairs. (1,1), (1,2), (1,3), (1,4), (1,5), (1,6) (2,1), (2,2), (2,3), (2,4), (2,5), (2,6) (3,1), (3,2), (3,3), (3,4), (3,5), (3,6) (4,1), (4,2), (4,3), (4,4), (4,5), (4,6) (5,1), (5,2), (5,3), (5,4), (5,5), (5,6) (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) 16 X = 5 X = 4 X = 2 X = 6 X = 7 X = 8 X = 9 X = 10 X = 11 X = 12

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The probability distribution for a random variable can be displayed in a table or a histogram. With a histogram, the probability of an event is the sum of the areas of the rectangles corresponding to the outcomes in the event. 18

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