The area under the curve of a Probability Density Function (PDF) represents probability.
Understanding the PDF and its Area
A Probability Density Function, often denoted as f(x), describes the likelihood of a continuous random variable taking on a given value. However, unlike probability mass functions for discrete variables, the value of the PDF f(x) itself at a specific point x does not directly represent a probability.
As the reference states: "A pdf f(x)... may give a value greater than one for some values of x, since it is not the value of f(x) but the area under the curve that represents probability."
This is a fundamental concept in continuous probability distributions:
- The value of the PDF f(x) at a point indicates the relative likelihood of the variable being near that point. A higher f(x) suggests a greater density of probability around x.
- The actual probability of the random variable falling within a specific range (say, between values a and b) is given by the area under the PDF curve between a and b.
Think of it like this:
| Concept | Represents |
|---|---|
| PDF Value f(x) | Relative likelihood or density at point x |
| Area Under Curve | Actual Probability over a range |
Key Takeaways
- The probability of a continuous variable taking on a specific value is effectively zero. Probability is meaningful only over an interval.
- The total area under the entire PDF curve must equal 1, representing the total probability of all possible outcomes.
- Calculating the area under the curve for a specific interval involves integration.
Therefore, when looking at a PDF graph, it's the accumulated area between two points on the x-axis that tells you the probability of the variable falling within that interval, not the height of the curve at any single point.
