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When we talk about the Non-negativity Property of a Probability Density Function (PDF), it means that the function must not produce negative values. Remember that probability, by definition, cannot be negative. A PDF, therefore, should be zero or positive everywhere over its entire domain. This is an essential property because probabilities represent the extent to which an event might occur. Negative probabilities are not meaningful in real-world scenarios.
In mathematical terms, if you have a PDF denoted as \( f(x) \), then for all possible values of \( x \), the function must satisfy \( f(x) \geq 0 \). This ensures that every outcome in the spectrum of the random variable has a sensible, non-negative association in terms of probability.

The Integral Equals One Property is another cornerstone of a valid PDF. This property states that if you add up the values of the PDF across its entire domain, the result should be equal to one. This is because the total probability of all possible outcomes of a random variable must sum to one. Probabilities are expressed as a fraction of one—where one represents certainty—and this integral property ensures that our PDF covers the entire spectrum of possibilities.
The mathematical expression that captures this property is:

• \( \int_{-\infty}^{\infty} f(x) \, dx = 1 \)

This equation guarantees that we don't miss any probability mass, ensuring the function accounts for all potential outcomes. This property is what differentiates PDFs from other types of functions.

A Random Variable is a fundamental concept in probability and statistics, representing a quantity whose outcome is subject to uncertainty. Unlike a fixed number, random variables can take different values, each with an associated probability. Imagine rolling a die—each face represents a potential value of a random variable.
Random variables come in two types:

• Discrete Random Variables: These take on a finite or countable set of possible outcomes, like the roll of a die.
• Continuous Random Variables: These can assume an infinite number of possible values within a given range, like measurements of time or height.

For continuous random variables, PDFs are used to determine the likelihood of the variable falling within a certain range. By understanding and using random variables, along with their distributions, we can model uncertainty in various real-life situations.