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The cumulative distribution function (CDF) is a fundamental concept in probability and statistics. It represents the probability that a random variable takes on a value less than or equal to a specific number. In mathematical terms, for a random variable \(X\), the CDF is denoted by \(F(x)\) and defined as:

• \(F(x) = P(X \leq x)\)

To find the CDF for a given probability density function (PDF), one integrates the PDF from its lower bound up to the variable of interest \(x\). This transforms the density of the data into a cumulative probability that is always non-decreasing and ranges from 0 to 1.

In our original exercise, we started with a PDF \(f(x) = 2x\), where \(0 \leq x \leq 2\). By integrating this, we determined that the CDF is \(F(x) = x^2\). This means the probability of the random variable being less than or equal to \(x\) is given by \(x^2\), within the specified interval.

A probability density function (PDF) is used to specify the probability of a continuous random variable falling within a particular range of values.
The key properties of a PDF are:

• The area under the entire PDF curve is equal to 1, which reflects the total probability of all possible outcomes.


• For any specific value, the PDF describes the relative likelihood of the random variable taking on that value.

Although the PDF represents probability theoretically, it must always be paired with calculations over intervals since the probability at any single point for a continuous variable is technically zero.
In the exercise, we observed \(f(x) = 2x\) as our PDF for \(x\) ranging from 0 to 2. This equation describes how probabilities are spread across this interval. The PDF gives a picture of where values are likely to appear, with the function increasing linearly, indicating higher probabilities as \(x\) increases.

The binomial coefficient is a key mathematical tool often used in statistics to solve problems involving discrete distributions. It is represented as \(\binom{n}{k}\) and pronounced as "n choose k". It calculates the number of ways to choose \(k\) successes from \(n\) possible trials.Here are some important characteristics:

• The binomial coefficient can be calculated using the formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where "!" denotes factorial.
• It plays a crucial role in combinatorics and is used extensively in binomial expansions, such as the binomial theorem.

In the step-by-step solution, we used the binomial coefficient when calculating the CDF for an order statistic. Specifically, it helped determine the coefficients in the formula by which we expanded our cumulative probability concerning the 12th order statistic. This expansion relied on the values \(\binom{13}{12}\) and \(\binom{13}{13}\).

Integration is a mathematical process used extensively in statistics, especially for continuous random variables. Integration allows us to calculate cumulative probabilities, expected values, and more.

Here are key integration applications in statistics:

• Calculating Cumulative Distribution Functions (CDF) from Probability Density Functions (PDF).
• Determining expected values, which is the average or mean outcome of a probability distribution, calculated as the integral over the entire range.
• Finding variances and higher moments that describe the distribution's characteristics.

In our exercise, integration was used to transform the PDF \(f(x)=2x\) into its corresponding CDF \(F(x) = x^2\). By integrating \(2x\) from 0 to \(x\), we obtained the cumulative probability, helping form the basis for calculating order statistics.