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A Discrete Uniform Distribution is a probability distribution where all outcomes are equally likely. Essentially, it means every number within a certain finite set has an equal chance of occurring. For example, when rolling a six-sided die, each side can appear with a probability of 1/6.

In our exercise, we consider the ranks of candidates for a job, ranging from 1 to \(n\). This distribution implies that each candidate has an equal chance of being selected, with a probability mass function (PMF) of \(p(x) = \frac{1}{n}\) for all ranks \(x\).

This concept is vital because it sets the groundwork for exploring properties like the expected value and variance of a random variable following this distribution.

The expected value, or mean, of a discrete random variable provides an average outcome if an experiment is repeated many times. In our scenario, it represents the average rank of a candidate when ranks are assigned randomly.

To calculate the expected value for a Discrete Uniform Distribution, you use the formula:

• \(E(X) = \sum x \cdot p(x)\).

For ranks from 1 to \(n\), this becomes

• \(E(X) = \sum\limits_{x=1}^{n} x \cdot \frac{1}{n} = \frac{1}{n} \sum\limits_{x=1}^{n} x\).

Using the formula for the sum of the first \(n\) positive integers, \(\sum\limits_{x=1}^{n} x = \frac{n(n+1)}{2}\), we find:

• \(E(X) = \frac{n+1}{2}\).

This result highlights that even in a random selection, there is a central tendency in rankings.

Variance measures how much the values of a random variable differ from the expected value. A higher variance indicates that the ranks are spread out over a wider range of values.

The variance \(V(X)\) for a discrete random variable is given by the shortcut formula:

• \( V(X) = E(X^2) - (E(X))^2 \).

First, calculate the expected value of the square of the ranks, \(E(X^2)\):

• \(E(X^2) = \frac{1}{n} \sum\limits_{x=1}^{n} x^2\).

By applying the formula for the sum of squares of the first \(n\) integers, \(\sum\limits_{x=1}^{n} x^2 = \frac{n(n+1)(2n+1)}{6}\), it becomes:

• \(E(X^2) = \frac{(n+1)(2n+1)}{6}\).

Then, substitute \(E(X)\) and \(E(X^2)\) into the variance formula resulting in:

• \(V(X) = \frac{n^2 - 1}{12}\).

This calculated variance determines the potential deviation of each rank from the expected average rank.

A Discrete Random Variable can take on a countable number of distinct values. It's a fundamental concept in probability theory that helps in analyzing quantitative outcomes, like rolling dice or ranking candidates in our exercise.

The uniqueness of a discrete random variable, such as our variable \(X\) which represents candidate ranks, lies in its probability distribution. Each possible value\(x\) has a probability \(p(x)\) associated with it, summing up to 1 for all possible outcomes.

Understanding discrete random variables allows us to predict and evaluate different statistical measures, such as expected value and variance, which describe the distribution of values neatly. These parameters are crucial for assessing uncertainties and making informed decisions based on probabilistic outcomes.