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The Probability Mass Function (PMF) is crucial in understanding discrete random variables, especially in cases like the hypergeometric distribution. In this exercise, the PMF pertains to the random variable \( X \), which represents the number of granite specimens selected from the 15 chosen specimens. The PMF provides the probability that \( X \) takes a specific value. Specifically, for the hypergeometric distribution used here, it gives the likelihood of selecting a particular number of granite specimens from the total available.
The PMF in hypergeometric distribution is calculated using the formula:

• \( P(X = k) = \frac{{\binom{10}{k} \binom{10}{15-k}}}{\binom{20}{15}} \)

where \( \binom{n}{k} \) represents combinations, reflecting the number of ways to choose \( k \) items from \( n \) items.
By understanding this PMF, students can determine the probability for any chosen number of granite specimens \( k \). This allows for precise calculations and insights into the behavior of the random variable within the dataset, as demonstrated in the exercise.

Standard deviation helps quantify the amount of variation or dispersion of a set of values. In the context of our hypergeometric distribution, it indicates how much the number of granite specimens selected is expected to vary from its mean under repeated sampling conditions.
For the exercise, the standard deviation \( \sigma \) of the number of granite specimens selected is evaluated using a specific formula for hypergeometric distributions:

• \( \sigma \approx \sqrt{15 \times \frac{10}{20} \times \frac{10}{20} \times \frac{5}{19}} \approx 1.58 \)

This indicates that on average, the number of granite specimens selected will usually deviate by about 1.58 from the mean of 7.5. Understanding standard deviation allows students to estimate the extent of variation around the mean value, providing insights such as the range of granite specimens typically selected. Additionally, it helps in understanding probability calculations, such as finding the probability that the selection is within one standard deviation of the mean.

A random variable is a fundamental concept in statistics, representing a numerical outcome of a random process. In the hypergeometric distribution exercise, the random variable \( X \) is defined as the number of granite specimens selected from the total of 15 specimens chosen.
Unlike fixed numbers, a random variable can take different values, each with a certain probability associated with it. In our exercise, these values are discrete, ranging usually from 5 to 10, as outlined by the problem conditions. Understanding \( X \) as a random variable allows students to grasp its probabilistic nature and its relationship with the PMF.
The random variable helps us to convey all possible outcomes of a statistical experiment in a structured way and is instrumental in the formation and use of probability mass functions, as well as calculating mean and standard deviation. Mastering the notion of a random variable is essential for engaging deeply with concepts in probability, statistics, and their practical application to exercises like this.