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In probability and statistics, a discrete uniform distribution refers to a situation where a finite number of outcomes are equally likely to occur. This means every possible outcome has the same probability of happening.

For example, consider a fair six-sided die. When rolled, each side (number) has a probability of 1/6 of showing up. In the exercise, the distribution includes outcomes of 2, 4, and 6, each appearing with a probability of 1/3.

The main characteristics of a discrete uniform distribution are:

• Equally likely outcomes
• A finite number of possible outcomes
• The sum of probabilities for all possible outcomes equals 1

Understanding this distribution helps to compute probabilities and statistics like the mean and variance, as it's evenly spread across its defined range.

The standard deviation is a statistical measure that represents the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

For the discrete uniform distribution in our example, we calculate the variance using the formula for a discrete uniform distribution, \[\sigma^{2} = \frac{(b-a+1)^{2}-1}{12}\]

where \(a\) and \(b\) are the minimum and maximum values of the distribution range, respectively. In this problem, the variance is 0.75.

To find the standard deviation \(\sigma\), simply take the square root of the variance, yielding \(\sigma = \sqrt{0.75} = 0.866\).

Standard deviation is crucial when assessing the distribution of data around its mean, particularly when calculating z-scores and probabilities.

A z-score indicates how many standard deviations an element is from the mean. It is a measure of how "off" a value is from what is expected based on the mean and standard deviation.

To calculate z-scores for the problem's sample means, we use the formula:\[z = \frac{X - \mu}{\sigma / \sqrt{n}}\]

where:

• \(X\) is the value of the sample mean
• \(\mu\) is the mean of the distribution
• \(\sigma\) is the standard deviation of the distribution
• \(n\) is the sample size

By calculating z-scores, we translate the values into standard normal form. This helps in determining the probability of a sample mean falling within a certain range. In this exercise, for \(x = 4.1\) and \(x = 4.4\), the z-scores are 1.33 and 2.51, respectively.

The sample mean represents the average of a sample, which is a subset of the entire population. It's calculated by summing all observed values in the sample set and dividing by the number of observations.

The sample mean is a crucial concept in statistics. It commonly serves as a point of estimate for the population mean. It's particularly useful in large datasets because it allows for predictions and analyses based on the subset's data.

In our problem, we are interested in the probability of the sample mean lying between two specific values: 4.1 and 4.4. When calculating probabilities and z-scores, the sample mean acts as a cornerstone, enabling the conversion between various statistical measures and the standard normal distribution. This is crucial in finding the relevant probabilities within a specified range.