The area under the probability density function (PDF) curve represents probability.
For a continuous random variable, unlike a discrete one, the probability of the variable taking on any single specific value is effectively zero. Probability is instead measured over intervals. The PDF, denoted as f(x), describes the likelihood of the variable falling within a particular range.
The crucial insight, as highlighted in the reference, is that:
"A pdf f(x), however, 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 means:
- The height of the PDF curve f(x) at a specific point x is a measure of probability density at that point, not the probability itself.
- To find the probability that the continuous random variable falls within a given range (say, between
aandb), you calculate the area under the curve of f(x) fromatob.
Understanding Area as Probability
Think of the total area under the entire PDF curve as representing the total possible outcome space. Since the variable must take on some value within its range, the total probability is 100%, and thus the total area under the PDF curve across all possible values of x is always equal to 1.
Key Aspects of the Area Under the PDF
Here are some important points about what the area under a PDF signifies:
- Probability Over an Interval: The area under the curve between two points,
aandb, gives the probability P(a ≤ X ≤ b) that the random variable X will take a value betweenaandb. - Total Probability: The area under the entire curve from the minimum possible value to the maximum possible value (often from negative infinity to positive infinity) is always exactly 1. This is a fundamental property of any valid PDF.
- Distinction from Discrete Variables: For discrete variables using a Probability Mass Function (PMF), the height of the bar at a specific value is the probability of that value. For continuous variables using a PDF, the height is just density; you need the area over a range to get probability.
Let's visualize this concept:
| Concept | Probability Mass Function (PMF) for Discrete Variables | Probability Density Function (PDF) for Continuous Variables |
|---|---|---|
| Probability at a Point | The height of the bar at x, P(X=x) | Zero (probability is defined over intervals) |
| Probability in a Range | Sum of the heights of bars within the range | Area under the curve within the range |
| What f(x) Represents | Probability of X=x | Probability density at x (not probability itself) |
| Total Probability | Sum of all f(x) equals 1 | Total Area under the curve equals 1 |
Practical Example
Imagine a PDF describing the lifespan of a certain type of lightbulb in hours.
- The height of the PDF curve at, say, 1000 hours (f(1000)) tells you the density of probability around 1000 hours, not the probability of a bulb lasting exactly 1000 hours (which is infinitesimally small).
- To find the probability that a lightbulb lasts between 900 and 1100 hours, you would calculate the area under the PDF curve between x=900 and x=1100. This area gives you P(900 ≤ Lifespan ≤ 1100).
In summary, the area under the PDF curve is the mechanism by which probabilities are quantified for continuous random variables, providing the probability that the variable falls within a specific range.
