7.2 Frequency and Probability Distributions

1. Frequency Distribution & Relative Frequency Distribution

2. Histogram of Probability Distribution3. Probability of an Event in Histogram 4. Random Variable5. Probability Distribution of a Random

Variable

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Frequency Distribution & Relative Frequency Distribution

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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.

Example Distributions

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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.

Example Distributions (2)

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Example Distributions

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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.

Histogram of Probability Distribution

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.

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Example Probability Distribution

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.

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Example Probability Distribution (2)

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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.

Probability of an Event in Histogram

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Example Probability of an Event

For the previous example, shade in the area that corresponds to the event "at least 3 heads."

Area is shaded in blue.

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Random Variable 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.

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Random Variable (2) 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 pi with p1 + p2 + … + pr = 1.

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Example Random Variable

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.

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Example Random Variable (2) There are equally likely

outcomes. X can be 0, 1, or 2.

10120

3N

8 8 23 2 17 7Pr( 0) , Pr( 1)10 1015 153 3

8 21 2 1Pr( 2)10 153

X X

X

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Example Random Variable & Distribution

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.

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X = 3

Example Random Variable & Distribution

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)

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X = 5X = 4

X = 2

X = 6X = 7X = 8X = 9X = 10X = 11X = 12

Example Random Variable & Distribution

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Summary Section 7.2 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.

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