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A probability density function (PDF) is a fundamental concept used to describe the distribution of continuous random variables. In simple terms, the PDF indicates the likelihood of a random variable taking on a particular value. However, unlike discrete probabilities, PDFs do not directly provide probabilities. Instead, the area under the curve of a PDF over a particular interval provides the probability that the random variable falls within that interval.

Key points to remember about PDFs include:

• The area under the entire PDF curve equals 1, representing total probability.
• The PDF curve's height can exceed 1 since it is not a probability measure by itself.
• PDFs are only applicable for continuous variables.

In our exercise, the triangular distribution is defined by a PDF consisting of two linear pieces, forming a peak at 50. This indicates that the value of 50 is the most likely to occur in this distribution.

```Integration in probability is a mathematical process used to find cumulative probabilities from a probability density function (PDF). By integrating the PDF over a specific interval, we can determine the probability that a random variable falls within this range.

Here are the essential points to understand when discussing integration in probability:

• Integration provides the area under the curve of the PDF, which corresponds to probability.
• It is particularly useful for continuous distributions, where probability is spread over a range of values.
• To find probabilities, you integrate from the lower bound to the upper bound of the desired interval.

In the exercise, integration was used to calculate:

• \(P(X \leq 40)\) by integrating from 30 to 40 for the function \(f(x) = 0.0025x - 0.075\)
• \(P(40 < X \leq 60)\) by integrating two portions: from 40 to 50 and from 50 to 60, with respective functions.```

Piecewise linear functions are mathematical functions composed of several linear segments. These segments are defined over specific intervals. Each segment has its unique linear equation. A key feature of piecewise linear functions is that they are not continuous in terms of slope, leading to sharp transitions at certain points.

Understanding piecewise linear functions involves:

• Identifying each linear segment and its corresponding domain (the interval over which it is defined).
• Noticing how segments fit together, typically meeting at so-called endpoints or breakpoints.
• Using these functions to model phenomena that change rates over different intervals.

In the triangular distribution provided in the exercise, the PDF is characterized by two piecewise linear functions:

• The increasing function \(f(x) = 0.0025x - 0.075\) for \(30 < x < 50\).
• The decreasing function \(f(x) = -0.0025x + 0.175\) for \(50 < x < 70\).

The piecewise nature captures the distinct linear rate changes in different segments of the distribution."