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The probability density function (pdf) is a crucial concept in statistics and probability. It represents the likelihood of a continuous random variable taking on a specific value. Understanding pdfs helps us model real-world scenarios where outcomes are not discrete but rather form a continuous spectrum.

For example, the pdf given in the exercise for the demand of propane gas describes how probable it is for the usage to be within a certain range. The formula is:

• \( f(x) = 2 \left( 1 - \frac{1}{x^2} \right) \) for \( 1 \leq x \leq 2 \)
• \( f(x) = 0 \) otherwise

This function tells us that amounts of gas between 1,000 and 2,000 gallons have variable probabilities, influenced by the shape of the curve (graph) defined by the function. The area under the curve of the pdf over its domain sums to 1, illustrating the total probability.

The expected value or mean of a random variable provides a measure of its central tendency. In simpler terms, it's the average outcome if an experiment is repeated many times.

We calculate the expected value of a continuous random variable by integrating the product of the variable's value and its pdf over the entire range:
\[E(X) = \int_{a}^{b} x \cdot f(x) \, dx\]
In our exercise, the expected demand for propane is calculated using:

• The integral \( \int_{1}^{2} x \cdot 2 \left( 1 - \frac{1}{x^2} \right) \, dx \)
• Simplifying the function within the integral before performing the integration

The outcome, after calculation, gives the expected amount of propane needed in a typical week.

Percentiles are statistical measures indicating the value below which a given percentage of observations fall. For continuous random variables, percentiles help us understand the distribution better by describing different points along the range.

For the \(100p\)th percentile, denoted as \ x_p \, we solve for \ x \ from the cumulative distribution function (CDF), where \( F(x) = p \).

In the exercise, this involves solving the equation:

• \( 2x - \frac{2}{x} = p \)
• Multiplying through by \( x \) and rearranging terms to form a quadratic equation: \( 2x^2 - px - 2 = 0 \)

The roots of this equation will provide the desired percentiles for different probabilities (\( p \)) in the context of propane demand.

Variance is a statistical measure indicating how much the values of a random variable deviate from the mean. It's a measure of dispersion or variability in the data.

To calculate the variance \( V(X) \), we first find \( E(X^2) \), the expected value of the square of the random variable:

• Integrate \( x^2 \cdot f(x) \) over the given range

Then, apply the formula:
\[V(X) = E(X^2) - [E(X)]^2\]
From our exercise, after determining \( E(X) \) and \( E(X^2) \), the variance informs us about the variability in weekly propane gas demand. This insight helps in planning and stocking the facility efficiently.

Integration is a fundamental tool in statistics, especially when dealing with continuous random variables. It helps us find aggregate quantities like totals and averages from continuous functions.

The integration process transforms the pdf into the cumulative distribution function (CDF) and calculates expected values, variances, and more.
The CDF is derived by integrating the pdf over its defined range:

• \( F(x) = \int_{lower}^{x} f(t) \), provides the probability that the random variable is less than or equal to \( x \)

In statistical analysis, this lets us compute probabilities and other essential metrics about an entire data set or distribution.
Understanding how to integrate functions is a key skill for interpreting statistical models, as evidenced by the various calculations performed in the propane exercise.