The probability density function (PDF) is a measure of continuous variables. It describes the relative likelihood of a continuous random variable taking on a given value.
Understanding the Probability Density Function
Unlike probability mass functions which apply to discrete variables and give the probability of a specific outcome, a PDF applies to variables that can take on any value within a continuous range.
Key characteristics:
- It does not directly give the probability of the variable taking on a specific value (which is zero for a continuous variable).
- Instead, the probability of the variable falling within a specific range of values is found by calculating the area under the curve of the PDF between the two points defining the range.
- The total area under the curve of a PDF over its entire range must equal 1, representing 100% probability.
Probability Density Function vs. Probability Mass Function
Here's a simple comparison:
| Feature | Probability Density Function (PDF) | Probability Mass Function (PMF) |
|---|---|---|
| Variable Type | Continuous Variables | Discrete Variables |
| Measures | Relative likelihood of values | Probability of specific values |
| Probability | Area under curve for a range | Direct value at a point |
Application in Finance
As highlighted in the reference, the probability density function measures continuous variables. While some real-world phenomena, like stock and investment returns, are technically discrete (prices change in specific increments), analysts often assume they are continuous.
This assumption is made for practical purposes:
- Modeling Performance: Continuous models using PDFs allow for smoother and more complex mathematical analysis.
- Analyzing Risks: Calculating probabilities of returns falling within certain ranges (e.g., potential losses or gains) is often simplified using continuous probability distributions.
Therefore, even when dealing with variables that are technically discrete, financial analysts may use PDFs by treating these variables as if they were continuous to facilitate analysis and modeling.
