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

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.

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.

 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

 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

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

 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

 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

 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

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

 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

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