Probability Density Function (PDF)

Definition

A probability density function (PDF) describes how likely different outcomes of a continuous random variable are. For any two values a and b, the probability that the variable falls between a and b equals the area under the PDF curve from a to b.

Key properties

• A PDF is non-negative everywhere.
• The total area under the PDF curve equals 1.
• PDFs are used to indicate which values are more or less likely for a continuous variable.

Why PDFs matter in finance

Financial analysts use PDFs to model the distribution of returns and to assess risk and expected outcomes. Although actual asset returns are technically discrete, analysts commonly assume continuous distributions to simplify modeling and risk analysis.

Interpreting PDF shapes

• Bell curve (normal distribution): Symmetric shape centered on the mean. A neutral risk profile is often implied when data resemble a normal curve.
• Skewness:
• Right skew (long tail to the right): Greater potential upside reward; the bulk of outcomes lie left of the mean.
• Left skew (long tail to the left): Greater downside risk; the bulk of outcomes lie right of the mean.
• Example property of a normal-like curve: about 68.5% of values fall within ±1 standard deviation of the mean.

Computation and visualization

Plotting a PDF can require calculus (differential equations or integrals) for analytic forms. In practice, statistical software or graphing tools are used to estimate and plot PDFs from data.

PDF vs. CDF

• PDF (probability density function): Shows the relative likelihood (density) of specific values.
• CDF (cumulative distribution function): Shows the probability that the variable is less than or equal to a given value; it is the cumulative area under the PDF up to that value and ranges from 0 to 1.

Relation to the Central Limit Theorem

The central limit theorem states that as sample size increases, the distribution of sample means tends to approach a normal distribution regardless of the original distribution’s shape. This explains why many aggregated or averaged financial measures are often modeled with normal-like PDFs.

Practical notes

• Investment returns are rarely perfectly normally distributed; empirical PDFs for financial data often show skewness and fat tails.
• Use PDFs as tools to compare likelihoods and quantify risk, but complement them with other measures (e.g., tail risk metrics) when assessing extreme outcomes.

Summary

A PDF provides a compact, visual way to understand how likely different outcomes are for a continuous variable. The shape of the PDF—its center, spread, skewness, and tails—communicates the probabilities of typical and extreme outcomes, making PDFs a fundamental concept in statistical analysis and financial risk assessment.