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Uniform Probability Distribution for Beginning Statistics

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Practice problems for these concepts can be found at:

Continuous Random Variables and Their Probability Distributions Solved Problems for Beginning Statistics

A continuous random variable is a random variable capable of assuming all the values in an interval or several intervals of real numbers. Because of the uncountable number of possible values, it is not possible to list all the values and their probabilities for a continuous random variable in a table as is true with a discrete random variable. The probability distribution for a continuous random variable is represented as the area under a curve called the probability density function, abbreviated pdf. A pdf is characterized by the following two basic properties: The graph of the pdf is never below the x axis and the total area under the pdf always equals 1.

The probability density function shown in Fig. 6-1 is a uniform probability distribution. This pdf represents the distribution of flight times between Omaha, Nebraska, and Memphis, Tennessee. The flight time is represented by the letter X. The graph shows that the flight times range from 90 to 100 minutes. The distance from the x axis to the graph remains constant at 0.10 and since the area of a rectangle is given by the length times the width, the area under the pdf is 10 × 0.10 = 1. Note that this pdf has the two basic properties given above. The graph of the pdf is never below the x axis and the total area under the pdf is equal to 1.

Uniform Probability Distribution

The representation of Fig. 6-1 by an equation is given as follows.

In general, if a random variable X is uniformly distributed over the interval from a to b, then the pdf is given by formula (6.1).

EXAMPLE 6.1 The probability that a flight takes between 92 and 97 minutes is represented as P(92 < X < 97) and is equal to the shaded area shown in Fig. 6-2. The rectangular shaded area has a width equal to 5 and a length equal to .1 and the area is equal to 5 × .1 = .5. That is, 50 percent of the flights between Omaha and Memphis will take between 92 and 97 minutes.

Mean And Standard Deviation For The Uniform Probability Distribution

The mean value for a random variable having a uniform probability distribution over the interval from a to b is given by

The variance for a uniform random variable is given by

EXAMPLE 6.2 The weights of 10-pound bags of potatoes packaged by Idaho Farms Inc. are uniformly distributed between 9.75 pounds and 10.75 pounds. The distribution of weights for these bags is shown in Fig. 6-3.

Mean And Standard Deviation For The Uniform Probability Distribution

Using formula (6.2), we see that the mean weight per bag is pounds and using formula (6.3), the standard deviation is If X represents the weight per bag, then P(X > 10.0) corresponds to the proportion of bags that weigh more than 10 pounds. This probability is shown in Fig. 6-4. The shaded rectangle has dimensions 1 and 0.75 and the area is 1 × 0.75 = 0.75. Seventy-five percent of the bags weigh more than 10 pounds. To help ensure consumer satisfaction, Idaho Farms instructs their employees to try and never underfill if possible. This is reflected in the average weight of 10.25 pounds per bag.

Mean And Standard Deviation For The Uniform Probability Distribution

The probability associated with a single value for a continuous random variable is always equal to zero, since there is no area associated with a single point. That is, the probability that X = a is given by

P(X = a) = 0

Practice problems for these concepts can be found at:

Continuous Random Variables and Their Probability Distributions Solved Problems for Beginning Statistics

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