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The mean of a discrete uniform distribution gives us the average or central value around which the data points (or random variables) are spread. In simpler terms, if you had a bunch of equally likely numbers, the mean is the middle of those numbers.

To calculate the mean for a discrete uniform distribution on a set of integers, you can use the formula:

• \( \mu = \frac{a + b}{2} \)

Here, \(a\) and \(b\) are the smallest and largest numbers in the set, respectively. For our example, if the numbers are from 0 to 99:

• \( a = 0 \)
• \( b = 99 \)

Plugging the values into the formula gives:

• \( \mu = \frac{0 + 99}{2} = 49.5 \)

So, the mean of our set of integers in this discrete uniform distribution is 49.5, nicely illustrating that the average, or the 'center', stands right between the minimum and maximum numbers in this evenly spread sequence.

Variance tells us how much the numbers in the distribution are spread out from the mean. A larger variance indicates a wider spread. For a discrete uniform distribution, the variance is calculated using a specific formula:

• \( \sigma^2 = \frac{(b-a+1)^2 - 1}{12} \)

In this formula, \(b\) is the largest integer, \(a\) is the smallest integer, and this expression gives us a measure of this spread of numbers. For example, using integers from 0 to 99:

• \( a = 0 \)
• \( b = 99 \)

Substituting these values yields:

• \( \sigma^2 = \frac{(99-0+1)^2 - 1}{12} = \frac{100^2 - 1}{12} = \frac{9999}{12} = 833.25 \)

So, the variance of our set is 833.25. This indicates that the numbers are widely spread around the mean (which we found as 49.5). The variance also tells us about the data's stability or variability in its distribution.

In a discrete uniform distribution, we've got a neat outcome: every possible value is equally likely. This is what makes it uniform. Each outcome, or value, has the same probability, calculated using:

• \( P(X=x) = \frac{1}{b-a+1} \)

It gives the probability of any specific integer \(x\) occurring in our range from \(a\) to \(b\). For instance, with numbers from 0 to 99:

• \( a = 0 \)
• \( b = 99 \)

The calculation of probability for any given integer becomes:

• \( P(X=x) = \frac{1}{99 - 0 +1} = \frac{1}{100} \)

This tells us that each of the 100 numbers is just as likely as any other, having a probability value of 0.01 (which is the same as 1%). This equality in probability makes discrete uniform distributions relatively easy to work with and predict.