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The Probability Density Function (PDF) provides the likelihood of a random variable falling within a particular range of values. The PDF is applicable only for continuous random variables and is an integral part of determining the probability of events within a specified interval.

For a function to be recognized as a legitimate PDF, it must integrate to 1 over the domain of the variable. In simpler terms, if you imagine the PDF as a curve over a graph, the total area under this curve must be equal to 1. This condition ensures that the sum of all probabilities across all possible outcomes equals certainty, which is 100% or 1.

In our specific example, the function is defined as:

• \(f(x) = 5(1-x)^4\) for \(0 < x < 1\)
• \(f(x) = 0\) otherwise

Here, \(f(x)\) describes a range of probabilities over the interval \(x\) from 0 to 1. To confirm this is a valid PDF, we compute the integral of \(5(1-x)^4\) from 0 to 1. Successfully computing this as 1 confirms its legitimacy, ensuring calculations based on this PDF will yield valid probabilities.

The Cumulative Distribution Function (CDF) is a critical concept when working with probability, especially in the context of continuous distributions. It represents the probability that a random variable \(X\) will take a value less than or equal to \(x\).

To find the CDF, we integrate the PDF over the range from the lowest bound up to a specified point \(x\). In our explained solution, the PDF \(f(x) = 5(1-x)^4\) is integrated as follows:

\[F(x) = \int_{0}^{x} 5(1-t)^4 dt = -(1-x)^5 + 1\]

• This function continuously increases as \(x\) increases from 0 to 1.
• At \(x=0\), the CDF is 0, and at \(x=1\), the CDF becomes 1, reflecting that the entire probability distribution is captured.

In any probability question, understanding the CDF helps assess the likelihood of different outcomes which can include the probability of a variable falling short of, or exceeding specific values.

Percentiles are a useful statistical measure that indicate the value below which a given percentage of data in a data set falls. When dealing with probability distributions, specifically continuous ones like ours, percentiles reflect key probabilities and potential cutoff points across the dataset.

To solve for a certain percentile, say the 99th percentile as in our example, we equate the CDF to this desired probability. So we solve for \(x\) in:

• \(F(x) = 0.99\)

This involves manipulating the CDF equation:

\(- (1-x)^5 + 1 = 0.99\)
Solving for \(x\) gives us the specific value (here \(x = 0.00623\)), which represents the maximum threshold under which 99% of data, or in this case, gas consumption, lies.

Understanding percentiles is valuable in real-life scenarios like determining tank capacity or inventory needs where you wish to avoid running out of resources with a certain confidence level. Thus, for this example, a tank capacity where only 1% risk of exhaustion exists corresponds to the value at the calculated 99th percentile.