What Is a Probability Density Function (PDF)?

What Is a Probability Density Function (PDF)?

Let me explain what a probability density function, or PDF, really is. It's an analytical tool that shows how likely different outcomes are across a continuous range of possibilities. In essence, the probability here represents the portion of a dataset's distribution that falls between two specific points. You'll often see financial analysts using PDFs to figure out how returns are spread out, which helps them assess the risks and expected values for investment prices and returns.

Key Takeaways

PDFs serve as a statistical measure to evaluate the chance that an investment's returns will land within a certain range of values. You can also use them to highlight the risks tied to a particular investment. Typically, these functions get plotted on graphs that look like bell curves, with all the data points sitting below the curve. If the curve skews at either end, that signals either greater or lesser risk and reward potential.

Understanding Probability Density Functions (PDFs)

In the investment world, people rely on statistical tools all the time. They're essential for analyzing market trends and figuring out the potential risks and returns of different investments. This kind of analysis lets investors and financial pros make smarter decisions about where to put their money.

The PDF itself measures how frequently investment returns fall into a specified range. You usually see it graphed, where a normal bell curve means neutral market risk, but a skewed curve points to higher or lower risk-reward scenarios.

Skewness happens when the taller part of the curve shifts left or right. If it's shifted left with a long tail on the right—that's right skew—it suggests greater upside reward. On the other hand, a right shift with a long left tail—left skew—indicates more downside risk.

Imagine normally distributed data on a bell curve graph. The mean is the central line, and vertical lines mark standard deviations, showing how far data points stray from the mean. The first two lines on either side cover about 68.5% of the data within plus or minus one standard deviation. For stock returns, this would mean 68.5% of the time, returns stay between those lines, with neutral risk and no skew.

Computing the PDF

To compute a PDF and plot it, you might need complex calculations involving hazard rates, differential equations, or integral calculus. In reality, you'll want to use graphing calculators or statistical software to handle this.

Remember, the PDF value can never be negative.

Example of a Probability Density Function (PDF)

PDFs deal with continuous variables, but stock and investment returns are usually discrete. Still, most analysts treat them as continuous to model performance and analyze risks.

Take the S&P 500 index over three years: when sequenced and plotted, it forms a bell curve with right skew, pointing to potential greater upside reward over that period.

Keep in mind, investment returns seldom follow a perfect normal distribution, so your graphs won't often be a clean bell curve.

What Does a Probability Density Function (PDF) Tell Us?

A PDF shows how likely certain outcomes are from a data process. It highlights which values are most probable compared to the rarer ones, and this varies based on the PDF's shape and features.

What Is the Central Limit Theorem (CLT) and How Does It Relate to PDFs?

The central limit theorem says that as your sample size grows, the distribution of a random variable approaches a normal distribution, no matter the original shape. For example, a single coin flip is binomial—heads or tails. But with many flips, say 10, getting five of each is most likely, while all heads is rare. Scale to 1,000 flips, and it looks like a normal bell curve. This ties into how PDFs model distributions in larger samples.

What Is a PDF vs. a CDF?

A PDF describes the likelihood of values appearing in a process at any point. A cumulative distribution function, or CDF, shows how those probabilities accumulate to 100% across all outcomes. With a CDF, you can determine the chance that a variable's outcome is less than or equal to a specific value.

The Bottom Line

Probability density functions describe the expected values of random variables from a sample. The PDF's shape tells you how likely observed values are. For stocks, prices and returns often follow a log-normal distribution, meaning downside losses happen more often than huge gains, compared to a normal distribution.