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The mean, or average, of a discrete uniform distribution is calculated by simply finding the midpoint between the smallest and largest values. This is because, in a discrete uniform distribution, every outcome is equally likely. This balance results in the mean being right in the center of the values.For example, if you have a PDF document where the number of pages ranges from 5 to 9, you will use 5 as your smallest value (\( a \)) and 9 as your largest value (\( b \)). The formula for finding the mean \( \mu \) is:\[ \mu = \frac{a + b}{2} \]Substituting the values from our scenario gives:\[ \mu = \frac{5 + 9}{2} = 7 \]This tells us that the average number of pages in our PDF document is 7, giving us an easy benchmark for what to expect in terms of document length.

The standard deviation measures how spread out the numbers in a distribution are from the mean. In a discrete uniform distribution, this calculation becomes straightforward with its specific formula. The standard deviation is crucial because it tells us how much variability we can expect.To calculate the standard deviation \( \sigma \), use the formula:\[ \sigma = \sqrt{\frac{(b-a+1)^2-1}{12}} \]Here, substituting the smallest value \( a = 5 \) and the largest value \( b = 9 \), we find:\[ \sigma = \sqrt{\frac{(9-5+1)^2-1}{12}} = \sqrt{\frac{5^2-1}{12}} = \sqrt{\frac{24}{12}} = \sqrt{2} \approx 1.41 \]This result means that most documents will have a number of pages within approximately 1.41 pages from the mean of 7. Understanding this helps you anticipate the diversity in document length.

A probability distribution provides a mathematical description of how the probabilities of a random variable are distributed. For a discrete uniform distribution, each outcome is equally likely, meaning each value has the same probability of occurring.In the context of our PDF document, where pages range from 5 to 9, there are five distinct possible outcomes. Each page count (5, 6, 7, 8, 9) has an identical probability since the distribution is uniform.Here’s how to understand the probability:- The total number of outcomes is 5.- The probability of any single outcome is computed by the formula:\[ P(x) = \frac{1}{n} \]where \( n \) is the number of outcomes. Therefore, for each count of pages in the document:\[ P(x) = \frac{1}{5} = 0.2 \]Thus, there is a 20% chance for each number of pages from 5 to 9. Recognizing this uniform spread helps in planning and understanding the random nature of document creation within these bounds.